Numerical Analysis

Algorithms in C

Version 4.2

 

 

 

 

 

 

 

 

 

 

User's Manual

For

"Numerical Analysis", fourth edition

Richard L. Burden and J. Douglas Faires

1988

 

 

 

 

 

Written by:

Harold A. Toomey, MSEE

Care-Free Software

3rd Quarter 1991

 

 

 

Technical Publications:

Harold A. Toomey

 

Programming:

Harold A. Toomey

 

 

© Copyright 1988-1993, Harold A. Toomey - All rights reserved

 

This document contains proprietary information of Harold A. Toomey and is protected by Federal copyright law.  The information may not be disclosed to third parties or copied or duplicated in any form, in whole or in part, without prior written consent of Harold A. Toomey.  Limited rights exist for individual and university site licenses.  The software may be used or copied only in accordance with the terms of the license agreement.  Students may copy this software with the intent to join the $20.00 Club, paying for the right to use this software.  See the sample license agreements in this document.

 

The information in this document is subject to change without notice.

 

 

"Numerical Analysis Algorithms in C" User's Manual

Version 4.2

Document Number 9307-42C-UM2

 

 

Care-Free Software

Attn: Harold Allen Toomey

1376 N. 1100 E.

American Fork, UT  84003

1-801-492-1526

 

 

IBM is a trademark of International Business Machines Corporation.

Microsoft and MS-DOS are registered trademarks.

UNIX is a registered trademark of AT&T Bell Laboratories.

VAX and VMS are registered trademarks of Digital Equipment Corporation.

 


This collection of C programs is dedicated to my wife, Holly and to my son, David.  They gave me the privacy I needed to program, and they listened attentively, sharing my enthusiasm, whenever I expounded on what I had programmed—even though they hadn't the foggiest idea what I was talking about.


 

 

 

PREFACE

 

 

About the Author

 

Harold A. Toomey, M.S. in Electrical and Computer Engineering, is currently a Software Engineer for Novell in Provo, Utah.  While minoring in mathematics at Brigham Young University, he tutored students in calculus, then tutored C programming at BYU's Electrical Engineering Department.  Not content with the provided FORTRAN algorithms while taking several numerical methods courses, he began coding numerical algorithms in C.  The introductory text used for these numerical analysis courses was "Numerical Analysis."

 

 

History of "Numerical Analysis Algorithms in C"

 

BYU's mathematics department expressed an interest in having all of the algorithms found in the "Numerical Analysis" text programmed in C, along with a few of their favorites still in FORTRAN.  Version 3.0 was finally completed in December 1988.  BYU was the first university to purchase a university site license.  This software is being used for their numerical methods courses today and has been tested by hundreds of students.  Their input has resulted in several other versions, culminating into version 4.2.  Version 4.0 became necessary for the fourth edition of the "Numerical Analysis" text.  Since 1988, several universities and scores of students have purchased these programs to be used in college course work and on the job.  See the file "revhist.doc" (for revision history) for a complete overview of the history of "Numerical Analysis Algorithms in C."

 

 

Acknowledgements

 

The author would like to express his appreciation to the many individuals who made suggestions for improvement on the previous versions of these algorithms.  These include the professor who gave directions for the first version: G. S. Gill, Brigham Young University (also a reviewer for the third edition of the text), and Bruce Cardwell who supervises the Numerical Analysis Laboratory also at Brigham Young University.  Special thanks also go to Jay Lawlor, M.S. Electrical Engineering, for giving timely feedback while using the algorithms for a numerical methods class at BYU.  In particular, thanks also goes to Holly Z. Toomey for typesetting previous versions of the Examples Book.

 


 

 

 

CONTENTS

 

PREFACE                ‑iv‑

 

1.    Introduction     1‑1

1.1     Getting Started         1‑1

1.2     Purpose of the Programs    1‑2

1.3     For Instructors          1‑2

1.3.1       "Numerical Analysis" Authors' Recommendations           1‑2

1.3.2       Homework Helpers            1‑2

1.3.3       Modifying Programs           1‑3

1.3.4       Intentionally Introducing Errors      1‑3

1.4     Product Support       1‑3

 

2.    Installation       2‑1

2.1     Basic Installation Procedures          2‑1

2.2     Uploading to Mainframe Computers            2‑2

 

3.    "Numerical Analysis Algorithms in C" Files     3‑1

3.1     Algorithm Files         3‑1

3.3     Supporting C Source Code              3‑5

3.4     Documentation Files            3‑6

3.5     Utility Files    3‑6

3.6     Batch, Script and Command Files              3‑7

3.7     File Structure Chart              3‑8

3.8     File Name Translation Table from 3rd to 4th Edition           3‑8

3.9     4th Edition Differences         3‑9

 

4.    Step-By-Step Examples on Various Computers         4‑1

4.1     Need List      4‑1

4.2     Customizing Naautil.c          4‑1

4.3     Example Using MS-DOS, Microsoft C and the P-Edit Editor          4‑2

4.4     Example Using UNIX, cc and the vi Editor   4‑5

4.5     Example Using a Macintosh and THINK C              4‑8

4.6     Example Using VAX/VMS, CC and the EDIT/EDT Editor    4‑12

 

5.    For Those New to C        5‑1

5.1     Mathematical Operators      5‑1

5.2     Mathematical Functions       5‑2

5.3     General Language Hints      5‑5

5.4     Language Transition Kit       5‑6

 

6.    Helps and Hints   6‑1

6.1     Generally Nice To Know      6‑1

6.1.1       Professor's Favorites, Must Have, Algorithms      6‑1

6.1.2       Homework Helper Algorithms        6‑1

6.1.3       Optional Title          6‑1

6.1.4       Optional File Saving           6‑2

6.1.5       Finding Functions              6‑2

6.1.6       Using Default Inputs          6‑2

6.1.7       Changing Arithmetic Precision      6‑2

6.1.8       Using Floating-Point Numbers in Functions          6‑3

6.1.9       The Pow() Function           6‑4

6.1.10     Implementing SIG-Digit Rounding/Truncation       6‑4

6.1.11     Floating-Point Output Alignment    6‑5

6.2     Converting Programs into Functions           6‑5

6.2.1       An Example Using Simpson's Rule          6‑7

6.3     Using Input Files (*.IN)         6‑8

6.4     Using Output Files (*.OUT)              6‑10

6.5     Explanation of the Naautil.c File       6‑10

6.5.1       #Define Flags        6‑10

6.5.2       Flag Default Settings         6‑11

6.5.3       Description of the Routines           6‑12

6.6     Using Naautil.c as Object Code       6‑14

6.6.1       MS-DOS    6‑15

6.6.2       UNIX           6‑15

6.6.3       Macintosh              6‑15

6.6.4       VAX/VMS    6‑16

6.7     Supporting C Source Code Usage List       6‑16

6.8     "Numerical Analysis" Text Errors and Corrections              6‑17

6.8.1       3rd Edition Errors              6‑17

6.8.2       4th Edition Errors   6‑18

6.9     Watch for These Run-Time Errors              6‑20

6.9.1       Stack Space          6‑20

6.9.2       Division By Zero     6‑20

6.9.3       Null Pointer Assignments              6‑20

6.9.4       No Disk Space       6‑21

6.9.5       Floating-Point Accuracy    6‑21

6.9.6       Program Stuck in an Infinite Loop             6‑21

 

7.    Useful Utilities     7‑1

7.1     Convert.c - Converting Files from Extended ASCII to Standard ASCII        7‑1

7.1.1       Why Convert.c is Needed             7‑1

7.1.2       How to Use Convert.c       7‑2

7.2     List.com - A Better TYPE Command          7‑3

7.3     Time-Saving Batch, Script and Command Files     7‑3

7.3.1       CC.BAT     7‑3

7.3.2       CCC           7‑5

7.3.3       VAXCC.COM         7‑6

 

8.    The Equation Evaluator Routines        8‑1

8.1     What the Routines Do         8‑1

8.2     How to Insert the Routines into a Program              8‑1

8.3     An Example Using Simpson's Rule             8‑2

8.4     Using Eqeval.c As Pre-Compiled Object Code       8‑2

8.5     Valid Math Operators and Functions           8‑3

8.6     Sample Equations    8‑4

8.7     Possible Error Messages    8‑4

8.8     List of Algorithms Using the Equation Evaluator Routines              8‑5

8.9     Limitations    8‑6

8.10   Trade-Offs    8‑6

 

9.    Portability        9‑1

9.1     C vs ANSI C             9‑2

9.2     IBM PCs and MS-DOS        9‑3

9.3     UNIX Workstations   9‑3

9.4     Macintosh Computers          9‑4

9.5     VAX Mainframes       9‑5

9.6     Tested Compilers    9‑5

 

10.  Sample License Agreements     10‑1

10.1   Individual License Sample   10‑1

10.2   University/Corporation Site License Sample           10‑3

 

11.  Packaging Information    11‑1

11.1   MS-DOS Diskettes              11‑1

11.1.1     5¼" 1.2M High Density Diskettes              11‑1

11.1.2     5¼" 360K Low Density Diskettes              11‑2

11.1.3     3½" 1.44M High Density Diskettes            11‑2

11.1.4     3½" 720K Low Density Diskettes              11‑2

11.2   Macintosh Diskettes             11‑2

11.2.1     3½" 800K Macintosh Diskettes     11‑3

 

12.  Purchasing Information              12‑1

12.1   $20.00 Club              12‑1

12.2   Order Form              12‑1

 

References             12‑2

 

Appendix A:           C Source Code for 041.C           A‑1

 

Appendix B:          C Source Code for NAAUTIL.C             B‑1

 

Appendix C:          Language Comparison Charts              C‑1

C.1    C vs Ada       C‑2

C.2    C vs BASIC              C‑8

C.3    C vs C++      C‑13

C.4    C vs FORTRAN 77              C‑14

C.5    C vs Pascal              C‑20

 

Appendix D:          Sample Programs in Other Languages            D‑1

D.1    Ada          D‑2

D.1.1      SIMPSON.ADA      D‑2

D.1.2      NAAUTIL.ADA        D‑4

D.1.3      SIMPSON.IN          D‑6

D.1.4      SIMPSON.OUT      D‑6

D.2    BASIC      D‑7

D.2.1      SIMPSON.BAS      D‑7

D.2.2      SIMPSON.IN          D‑8

D.2.3      SIMPSON.OUT      D‑9

D.3    C              D‑10

D.3.1      SIMPSON.C           D‑10

D.3.2      NAAUTIL.H             D‑11

D.3.3      SIMPSON.IN          D‑14

D.3.4      SIMPSON.OUT      D‑14

D.4    C++         D‑15

D.4.1      SIMPSON.CPP      D‑15

D.4.2      NAAUTIL.HPP        D‑16

D.4.3      SIMPSON.IN          D‑18

D.4.4      SIMPSON.OUT      D‑18

D.5    FORTRAN 77           D‑19

D.5.1      SIMPSON.FOR     D‑19

D.5.2      SIMPSON.IN          D‑21

D.5.3      SIMPSON.OUT      D‑21

D.6    Pascal     D‑22

D.6.1      SIMPSON.PAS      D‑22

D.6.2      NAAUTIL.INC         D‑24

D.6.3      NAAMATH.INC       D‑25

D.6.4      SIMPSON.IN          D‑25

D.6.5      SIMPSON.OUT      D‑25

 

 



 

 

 

1.  Introduction

 

 

"Numerical Analysis Algorithms in C" contains 116 stand-alone programs implementing the algorithms found in the texts:

 

"Numerical Analysis", third and fourth edition,

Richard L. Burden & J. Douglas Faires, 1988.

 

Each program is written in ANSI C to make them more portable to other computer systems.  They should run on any computer with a reasonable C compiler, such as IBM PCs, UNIX workstations, VAXes, and Macintoshes.

 

The "Numerical Analysis" text, hereafter referred to as "the text", covers the following numerical topics:

 

          Chapter 1   - Mathematical Preliminaries

          Chapter 2   - Solutions of equations in one variable

          Chapter 3   - Interpolation and polynomial approximation

          Chapter 4   - Numerical differentiation and integration

          Chapter 5   - Initial-value problems for ordinary differential equations

          Chapter 6   - Direct methods for solving linear systems

          Chapter 7   - Iterative techniques in matrix algebra

          Chapter 8   - Approximation theory

          Chapter 9   - Approximating eigenvalues

          Chapter 10 - Numerical solutions of nonlinear systems of equations

          Chapter 11 - Boundary-value problems for ordinary differential equations

          Chapter 12 - Numerical solutions to partial differential equations

 

From these topics, "Numerical Analysis Algorithms in C" has programmed routines for: vector and matrix manipulation, linear equations (LU decomposition/backsolving, matrix inversion, etc.), matrix/vector norms, eigenvalue/vectors, complex number and polynomial manipulation, least-square polynomial approximation, FFTs, numerical integration, root finding, solution of nonlinear equations, Taylor polynomial approximation, cubic splines, derivatives, ordinary and partial differentiation.

 

This User's Manual will help you to use these programs to their fullest potential.  It will walk you through an example, tutor you if you are unfamiliar with the C language, introduce you to several useful utilities, and assist you when running these programs on different computer systems.

 

 

1.1 Getting Started

 

To install "Numerical Analysis Algorithms in C" onto your computer system, see Chapter 2 - "Installation."  If you are new to the C programming language, you may wish to read through Chapter 5 - "For Those New to C."  If you want a detailed example using various C compilers and operating systems, see Chapter 4 - "Step-By-Step Examples on Various Computer Systems."

 

This software package contains about 1.5M bytes of files.  If disk space is limited, then just copy the eight supporting ".c" files ("complex.c", "eqeval.c", "gaussj.c", "naautil.c", "naautil2.c", "naautil3.c", "round.c" and "trunc.c") and the desired algorithms onto your disk.  The eight supporting files require about 100K of disk space.  If you are running these algorithms from a floppy disk, be sure to leave the write protect tab off so the programs can save their output to a file.  If this is undesirable, see Sub-Section 6.1.4 - "Optional File Saving."

 

If you feel comfortable with C, go ahead and compile and run an algorithm.  The source code is very readable and user friendly.  To see what the algorithm numbers correspond to, see Section 3.1 - "Algorithm Files."  This is the most important list in this manual and should be printed out for frequent reference.  Section 3.1 is also given in the file "readme.doc" for your convenience.

 

 

1.2 Purpose of the Programs

 

These programs are fast, but are not optimized for speed.  As stated by the authors in the text's preface:

 

"Although the algorithms will lead to correct programs for the examples and exercises in the text, it must be emphasized that there has been no attempt to write general-purpose software.  In particular, the algorithms have not always been listed in the form that leads to the most efficient program in terms of either time or storage requirements."

 

The purpose of these programs is to teach students numerical methods, not programming and optimization skills.  For a good book of general-purpose mathematical software, see the book "Numerical Recipes in C" listed in the references.  These programs can also be used as a tool for building other programs.  Once the algorithms are understood, they can be more easily enhanced for general-purpose applications.

 

 

1.3 For Instructors

 

This software package is intended to be used by instructors of numerical methods/analysis courses.  The best way to learn numerical methods is to program the algorithms from scratch and have them run on a computer.  This is a time consuming process and may take a "good" programmer from 1 to 5 hours per program.  Students can best benefit from these programs AFTER taking the appropriate numerical analysis courses.

 

 

1.3.1        "Numerical Analysis" Authors' Recommendations

 

The authors of the text "Numerical Analysis" mention in the preface that:

 

"Actual programs are not included because, in our experience, this encourages some students to generate results without fully understanding the method involved."

 

In other words, as an instructor, you may consider giving your students only selected main algorithms, and definitely not the "Homework Helpers" algorithms as discussed below.

 

 

1.3.2        Homework Helpers

 

Roughly half of the included programs are labeled as "Homework Helpers."  Most of these programs modify the given text algorithms to satisfy the homework exercises in the text.  An example of this is turning Algorithm 2.4 - Secant Method ("024.c") into the Method of False Position ("024B.c").  Use these "homework helpers" to correct homework assignments.  Do NOT just give these out to your students.  Most modifications will take only a short time to implement, once the algorithm is understood.

 

 

1.3.3        Modifying Programs

 

These algorithms are given as a learning tool.  Modifying them is part of the learning process.  These algorithms may be modified by the instructor or by the students, even though this package is copyrighted.  They may not, however, be altered to be resold for profit without prior written consent from the programmer.  See the sample licensing agreements in Chapter 10 for more details.

 

 

1.3.4        Intentionally Introducing Errors

 

As an alternative to withholding these programs from your students, you may wish to give them a copy with intentionally introduced errors.  This would cause them to search the entire program over for correctness, bridging the gap between giving too little or too much information.

 

 

1.4 Product Support

 

If questions arise, ranging from getting these algorithms to work with your compiler to adapting a particular algorithm to a specific application, just call CARE-FREE SOFTWARE at 1-801-785-0464.  The programmer will answer your questions at no charge other than the normal phone charges on your monthly phone statement.  Enhancements, recommendations and bug reports are always welcomed.



 

 

 

2.  Installation

 

 

2.1 Basic Installation Procedures

 

 

The "Numerical Analysis Algorithms in C" programs do not come with an installation program.  To install these algorithms onto your computer, do the following steps:

 

1.       Make a set of backup diskettes.  See your operating system manual for specifics.

 

2.       Make another set of "working" diskettes or copy the diskettes onto your hard disk.  All 500+ files combined require less than 1.5M bytes of disk space.  Only a couple of the files are required at a time to get the algorithms to work properly, making them useful even on systems without a hard disk.

 

3.       You may want to convert each file on the "working" disk from extended ASCII to standard ASCII.  This is usually required for Macintoshes, most UNIX computers, and VAXes.  Failure to do so may result in scrambled looking output characters.  Use "convert.exe", as explained in Section 7.1, to do this task relatively easily.  Macintosh disks ordered from Care-Free Software have had this step done already.

 

4.       It is recommended that the algorithms be placed in their own sub-directory (or Macintosh folder), such as "naa42."  This sub-directory can be created and entered by typing one of the following sets of commands:

 

          MS-DOS:

                   C:\> MD NAA42                                   - make directory

                   C:\> CD NAA42                                   - change directory

                   C:\NAA42> DIR /P                           - show directory contents

 

          UNIX:

                   % mkdir naa42                                   - make directory

                   % chdir naa42                                   - change directory

                   % pwd                                                       - show current directory

                   % ls -alF                                             - show directory contents

 

          VAX/VMS:

                   $ CREATE/DIR [SMITH.NAA42]  - make directory

                   $ SET DEFAULT [.NAA42]            - change directory

                   $ SHOW DEFAULT                                - show current directory

                   $ DIR/SIZE/DATE                              - show directory contents

 

5.       To be able to run every program from a floppy diskette, eight support files are required:

 

        complex.c      naautil.c      round.c

        eqeval.c       naautil2.c     trunc.c

        gaussj.c    naautil3.c 

 

These files require about 100K bytes of disk space.  The desired algorithm files such as "041.c" are also needed.  The majority of the algorithms need only "naautil.c" which is about 20K bytes large.

 

6.       If the programs do not compile correctly, you may need to change some flags inside the "naautil.c" file.  Use your text editor to modify this file.  The contents of "naautil.c" should be self-documenting.  These flags are defined near the top of the file.  See Section 6.5 - "Explanation of the Naautil.c File" if more detailed information is desired.

 

In the event that nothing seems to be working, you can set both the EQ_EVAL and the FILE_SAVE flags to FALSE.  This will disable the options to save the output to a file and to use the Equation Evaluator routines, but the algorithms will usually work.  These two options use variable length argument lists, which may not work on older compilers.

 

7.       If all else fails, ask another C programmer for help or call CARE-FREE SOFTWARE for free technical support.

 

 

2.2 Uploading to Mainframe Computers

 

To get these programs onto many workstations or mainframe computers, communications software is usually required.  A well-supported communications protocol is known as Kermit.  An example using Kermit looks something like this:

 

NOTE:        This example uses CALL/ProComm to transfer files onto a VAX/UNIX workstation.

 

1.       Log onto the mainframe using CALL, ProComm or your favorite communications package.  Select kermit as the transfer protocol.  Use binary mode to send files containing extended ASCII characters.  Use ASCII mode if the files have been converted to standard ASCII by the "convert.exe" program.  Binary mode is slower than ASCII mode.  Remember that C files are case sensitive.

 

2.       On the mainframe, change to an appropriate directory and type:

 

For a VAX, type:

$ use kermit                                               (Do NOT type "$ kermit")

Kermit-32> set file_type binary         (or:  set file_type ascii)

Kermit-32> receive

 

For a UNIX workstation, type:

% kermit

Kermit-32> set binary                        (or:  set ascii)

Kermit-32> receive

 

3.       On your PC, immediately issue the file sending commands.

 

For CALL, type:

[F9] File Send Kermit

File to transfer: filename

 

For ProComm, type:

[ALT] K

                   2) Send

Please enter filespec: filename

 

4.       Patiently wait as the file(s) are transferred to the mainframe.  The use of wild cards is recommended (ie - *.C instead of filename).

 

5.       Exit kermit on the mainframe.

Kermit-32> exit

$ logout

 

A host full of other issues have been left to the user, such as baud rate, parity, stop bits, duplex, use of wild cards, etc.  These are unique to each computer system and communications software package.

 

You may want to convert the files from extended ASCII to standard ASCII (using "convert.c") before uploading them to a mainframe computer.  If you plan to view and print your work on an IBM PC but compile and run the algorithms on a mainframe, you may want to keep the files in extended ASCII.

 

Test your preferences using Algorithm 4.1 ("041.c").  It uses three different extended ASCII characters to form an integral sign: '!', '#' and '"'.  "Convert.c" changes these three characters into standard ASCII: '[', '|' and ']'.



 

 

 

3.  "Numerical Analysis Algorithms in C" Files

 

 

This software package contains 116 algorithms.  Each algorithm has been coded as a stand-alone program.  Each program prompts for input, executes the algorithm as described in the text "Numerical Analysis", and prints the results.  Other math packages provide only subroutines, requiring a programmer to insert them inside a program and either hard code or prompt for the inputs and print the outputs.

 

The files are catagorized as follows, where "nnn" represent algorithm numbers like "041" for Algorithm 4.1:

 

a.       nnn.C          Algorithms from the text "Numerical Analysis" fourth edition.  (57 total)

 

b.       nnnA.C       Algorithms not found in the text.  Included as "Professor Favorites, Must Have" as recommended by mathematics professors at Brigham Young University.  (6 total)

 

c.       nnnB.C, nnnC.C, and nnnD.C

Algorithms included as "Homework Helpers."  Some are asked for in the homework exercises while others are for helping with important concepts covered in the text.  These can save hours of coding on the homework exercises.  (53 total)

 

d.       *.C              NAA supporting files containing 57 functions.  (8 total)

 

e.       *.IN              Input files used to test each algorithm.  They match the inputs to the example problems presented after each algorithm in the text.  (116 total)

 

f.        *.OUT         Output files used to test each algorithm.  They match the outputs to the example problems presented after each algorithm in the text.  (116 total)

 

g.       *.EXE                   Executable programs for each algorithm.  The default functions (like f(x)) are the same as those used in the example problems presented after each algorithm in the text.  These programs must be purchased separately and are currently available only for MS-DOS and Macintosh computers.  (116 total)

 

h.       *.DOC        Documentation in simple text file format.  Includes "readme.doc", "revhist.doc" and "usersman.doc."

 

Each program was tested on the sample problems given in the text just after the algorithm description.  These sample solutions are found in the OUT sub-directory in files named with a ".out" extension.  Their inputs are found in the IN sub-directory in files named with a ".in" extension.

 

Over two-thirds of the algorithms need to be compiled only once.  They are marked with an asterisk (*) on the table below.  Of these algorithms, nearly half are able to prompt you for an equation during run-time.  See Chapter 8 - "The Equation Evaluator Routines" for more details.

 

 

3.1 Algorithm Files

 

 

CHAPTER 1      Mathematical Preliminaries

 

COMPLEX.C     - "Numerical Recipes in C" Complex Number Routines

EQEVAL.C        - Equation Evaluator Routines

GAUSSJ.C         - "Numerical Recipes in C" Gauss-Jordan Matrix Solver

NAAUTIL.C        - "Numerical Analysis Algorithms in C" Utilities I             (standard)

NAAUTIL2.C      - "Numerical Analysis Algorithms in C" Utilities II            (extended)

NAAUTIL3.C      - "Numerical Analysis Algorithms in C" Utilities III            (complex)

ROUND.C         - Rounds a floating-point value to SIG significant digits

TRUNC.C          - Truncates a floating-point value to SIG significant digits

011B.C*             - Taylor Polynomial Approximation                                                     Algorithm 1.1B

 

 

CHAPTER 2      Solutions of Equations in One Variable

 

021.C*               -  Bisection (or Binary-Search)                                        Algorithm 2.1

022.C*               -  Fixed-Point                                                                                       Algorithm 2.2

023.C                 -  Newton-Raphson                                                                             Algorithm 2.3

024.C*               -  Secant                                                                                             Algorithm 2.4

024B.C*             -  Method of False Position (or Regula Falsi)                                     Algorithm 2.4B

024C.C              -  Modified Newton-Raphson Method                                                  Algorithm 2.4C

025.C*               -  Steffensen                                                                                       Algorithm 2.5

026.C*               -  Horner                                                                                              Algorithm 2.6

027.C*               -  Müller                                                                                               Algorithm 2.7

028A.C*             +  Complex Polynomial Solver (CPOLY)                                           Algorithm 2.8A

 

 

CHAPTER 3      Interpolation and Polynomial Approximation

 

031.C*               -  Neville's Iterated Interpolation                                                          Algorithm 3.1

031B.C*             -  Neville's Iterated Interpolation (with rounding)                                 Algorithm 3.1B

031C.C*             -  Aitken's Iterated Interpolation                                        Algorithm 3.1C

032.C*               -  Newton's Interpolatory Divided-Difference Formula    Algorithm 3.2

033.C*               -  Hermite Interpolation                                                                        Algorithm 3.3

034.C*               -  Natural Cubic Spline                                                                        Algorithm 3.4

035.C*               -  Clamped Cubic Spline                                                                    Algorithm 3.5

 

 

CHAPTER 4      Numerical Differentiation and Integration

 

040B1.C            -  1st Derivative Approximation (for functions)                                   Algorithm 4.0B1

040B2.C*           -  1st Derivative Approximation (for tabulated data)        Algorithm 4.0B2

040B3.C            -  1st Derivative Approximation (for functions w/TOL)    Algorithm 4.0B3

040C1.C            -  2nd Derivative Approximation (for functions)                                  Algorithm 4.0C1

040C2.C*           -  2nd Derivative Approximation (for tabulated data)       Algorithm 4.0C2

040D1.C*           -  Richardson's Extrapolation                                                             Algorithm 4.0D1

040D2.C*           -  Richardson's Extrapolation (with rounding)                                    Algorithm 4.0D2

041.C*               -  Composite Simpson's Rule                                                            Algorithm 4.1

041B.C*             -  Composite Trapezoidal Rule                                       Algorithm 4.1B

041C.C*             -  Composite Midpoint Rule                                                                Algorithm 4.1C

041D.C*             -  Newton-Cotes Formulas for Integrals (8 total)                                Algorithm 4.1D

042.C*               -  Adaptive Quadrature                                                                        Algorithm 4.2

043.C*               -  Romberg                                                                                          Algorithm 4.3

043B.C*             -  Gaussian Quadrature                                                                      Algorithm 4.3B

044.C                 -  Composite Simpson's Rule for Double Integrals                            Algorithm 4.4

044B.C              -  Composite Trapezoid Rule for Double Integrals                             Algorithm 4.4B

044C.C              -  Gaussian Quadrature for Double Integrals                                     Algorithm 4.4C

045.C                 -  Composite Simpson's Rule for Triple Integrals                              Algorithm 4.5

045B.C              -  Composite Trapezoid Rule for Triple Integrals                               Algorithm 4.5B

045C.C              -  Gaussian Quadrature for Triple Integrals                     Algorithm 4.5C

 

 

CHAPTER 5      Initial-Value Problems for Ordinary Differential Equations

 

051.C*               -  Euler                                                                                                                 Algorithm 5.1

051B.C*             -  Midpoint, Modified Euler, and Heun's Methods                                Algorithm 5.1B

052.C*               -  Runge-Kutta (Order Four)                                                               Algorithm 5.2

053.C                 -  Runge-Kutta-Fehlberg                                                                     Algorithm 5.3

054.C*               -  Adam's Fourth-Order Predictor-Corrector                                      Algorithm 5.4

054B.C*             -  Adams-Bashforth (all four) and Milne's Methods                            Algorithm 5.4B

054C.C*             -  Milne-Simpson Predictor-Corrector                                                Algorithm 5.4C

055.C*               -  Adam's Variable Step-Size Predictor-Corrector                             Algorithm 5.5

056.C*               +  Extrapolation                                                                                   Algorithm 5.6

057.C                 -  Runge-Kutta for Systems of Differential Equations                        Algorithm 5.7

057B.C              -  Euler's Variable Step-Size for Systems                                          Algorithm 5.7B

058.C                 -  Trapezoidal with Newton Iteration                                                   Algorithm 5.8

 

 

CHAPTER 6      Direct Methods for Solving Linear Systems

 

060B.C*             -  Matrix Inverter                                                                                  Algorithm 6.0B

060C.C*             -  Determinant of a Matrix                                                                   Algorithm 6.0C

060D.C*             -  Matrix Multiplier                                                                                Algorithm 6.0D

061.C*               -  Gaussian Elimination with Backward Substitution                          Algorithm 6.1

061B.C*             -  Gaussian Elimination with Backward Substitution                          Algorithm 6.1B 

                              (with rounding)

061C1.C*           - Gauss-Jordan Method                                                                      Algorithm 6.1C1

061C2.C*           - Gauss-Jordan Method (with rounding)                          Algorithm 6.1C2

061D1.C*           - Gaussian-Elimination - Gauss-Jordan Hybrid Method  Algorithm 6.1D1

061D2.C*           - Gaussian-Elimination - Gauss-Jordan Hybrid Method  Algorithm 6.1D2

                             (with rounding)

062.C*               -  Gaussian Elimination with Maximal Column Pivoting  Algorithm 6.2

062B.C*             -  Gaussian Elimination with Maximal Column Pivoting  Algorithm 6.2B

                              (with rounding)

063.C*               -  Gaussian Elimination with Scaled Column Pivoting    Algorithm 6.3

063B.C*             -  Gaussian Elimination with Scaled Column Pivoting    Algorithm 6.3B

                              (with rounding)

064.C*               -  Direct Factorization                                                                         Algorithm 6.4

064B.C*             -  Direct Factorization which solves AX=B                      Algorithm 6.4B

064C.C*             -  Direct Factorization with Maximal Column Pivoting     Algorithm 6.4C

                              (3rd edition)

065.C*               -  LDLt Factorization                                                                           Algorithm 6.5

065B.C*             -  LDLt Factorization which solves AX=B                                           Algorithm 6.5B

066.C*               -  Choleski                                                                                           Algorithm 6.6

066B.C*             -  Choleski which solves AX=B                                       Algorithm 6.6B

067.C*               -  Crout Reduction for Tridiagonal Linear Systems                            Algorithm 6.7

 

 

CHAPTER 7      Iterative Techniques in Matrix Algebra

 

070B.C*             -  Vector and Matrix Norms                                                                 Algorithm 7.0B

071.C*               -  Jacobi Iterative                                                                                 Algorithm 7.1

072.C*               -  Gauss-Seidel Iterative                                                                     Algorithm 7.2

073.C*               -  Successive Over Relaxation (SOR)                                               Algorithm 7.3

074.C*               -  Iterative Refinement (with rounding)                                                Algorithm 7.4

074B.C*             -  Iterative Refinement (single-precision)                        Algorithm 7.4B

 

 

CHAPTER 8      Approximation Theory

 

080B.C*             -  Least-Squares Polynomial Approximation                                      Algorithm 8.0B

081.C*               +  Fast Fourier Transformation                                       Algorithm 8.1

 

 

CHAPTER 9      Approximating Eigenvalues

 

091.C*               -  Power Method                                                                                  Algorithm 9.1

091B.C*             -  Power Method with Aitken's Delta2 Method                                     Algorithm 9.1B

092.C*               -  Symmetric Power Method                                                               Algorithm 9.2

093.C*               -  Inverse Power Method                                                                     Algorithm 9.3

094.C*               -  Wielandt's Deflation                                                                         Algorithm 9.4

094B.C*             -  Wielandt's Deflation using Power Method for lambda1                  Algorithm 9.4B

O095.C*            -  Householder Method                                                                        Algorithm 9.5

095B.C*             -  Householder Method (3rd edition)                                                   Algorithm 9.5B

095C.C*             -  Householder Method for Non-Symmetric Matrices      Algorithm 9.5C

                              (Upper Hessenberg)

095D.C*             -  Householder Method (with rounding)                                               Algorithm 9.5D

096.C*               -  QR Algorithm                                                                                   Algorithm 9.6

096B.C*             -  QL Algorithm (3rd edition)                                                               Algorithm 9.6B

 

 

CHAPTER 10    Numerical Solutions of Nonlinear Systems of Equations

 

101.C                 -  Newton's Method for Systems                                                        Algorithm 10.1

101A.C               -  Steffensen's Method for Systems                                                   Algorithm 10.1A

102.C                 -  Broyden's Method for Systems                                                       Algorithm 10.2

103.C                 -  Steepest Descent Method (with F(x) and J(x))                               Algorithm 10.3

103B.C              -  Steepest Descent Method (with G(x) and gradG(x))   Algorithm 10.3B

 

 

CHAPTER 11    Boundary-Value Problems for Ordinary Differential Equations

 

111.C                 -  Linear Shooting                                                                                Algorithm 11.1

112.C                 -  Nonlinear Shooting with Newton's Method                                      Algorithm 11.2

112B.C              -  Nonlinear Shooting with Secant Method                      Algorithm 11.2B

113.C                 -  Linear Finite Difference                                                                   Algorithm 11.3

113B.C              -  Linear Finite Difference (Richardson's Extrapolation) Algorithm 11.3B

114.C                 -  Nonlinear Finite Difference                                                              Algorithm 11.4

114B.C              -  Nonlinear Finite Difference (Richardson's Extrapolation)               Algorithm 11.4B

115.C                 -  Piecewise Linear Rayleigh-Ritz                                                      Algorithm 11.5

116.C                 -  Cubic Spline Rayleigh-Ritz                                                              Algorithm 11.6

 

 

CHAPTER 12    Numerical Solutions to Partial-Differential Equations

 

121.C                 -  Poisson Equation Finite-Difference (Elliptic)                                   Algorithm 12.1

122.C*               -  Heat Equation Backward-Difference (Parabolic)                            Algorithm 12.2

122B.C*             -  Heat Equation Forward-Difference (Parabolic)                               Algorithm 12.2B

122C.C*             -  Heat Equation Richardson's Method (Parabolic)                            Algorithm 12.2C

123.C*               -  Crank-Nicolson (Parabolic)                                          Algorithm 12.3

124.C                 -  Wave Equation Finite-Difference (Hyperbolic)                                Algorithm 12.4

125.C                 -  Finite-Element                                                                                 Algorithm 12.5

126A.C               -  Parabolic Equations With Newton Iteration in 1-D       Algorithm 12.6A

127A.C               -  Parabolic Equations With Newton Iteration in 2-D       Algorithm 12.7A

128A.C               -  Elliptic Equations With Newton Iteration in 2-D                               Algorithm 12.8A

129A.C               -  Biharmonic Equation Using Gauss-Jordan Method     Algorithm 12.9A

 

The '+'s above mean the program may need a larger stack when compiled and linked.

The '*'s above mean the program needs to be compiled only once.

 

 

3.3 Supporting C Source Code

 

The eight files below are needed to compile each and every program.  Most algorithms require only one or two of them at a time.

 

COMPLEX.C

"Complex.c" contain several routines for operating on complex numbers.  It originated from the book "Numerical Recipes in C" and is only used in "naautil3.c."

 

EQEVAL.C

"Eqeval.c" contains the Equation Evaluator routines.  These routines enable a program to enter and evaluate an equation during run-time. It is useful within algorithms that need to evaluate a single function such as f(x) or f(y,t).  It is used by 34 algorithms.  See Chapter 8 - "The Equation Evaluator Routines" for more details on this file.

 

GAUSSJ.C

"Gaussj.c" is a Gauss-Jordan matrix solver routine.  It originated from the book "Numerical Recipes in C."  It is used by only 9 of the algorithms.

 

NAAUTIL.C

"Naautil.c" contain important routines used by all of the algorithms.  Most are for dynamically allocating memory for arrays.  Some of the routines originated from the book "Numerical Recipes in C."  See Section 6.5 - "Explanation of the Naautil.c File."

 

NAAUTIL2.C

"Naautil2.c" contains more dynamically allocated memory routines for less-used data types.  it is used only 2 times.

 

NAAUTIL3.C

"Naautil3.c" contains more dynamically allocated memory routines for complex data types.  It is used only 3 times.

 

ROUND.C

"Round.c" rounds a floating-point value to SIG significant digits.  Only 9 algorithms currently use this function.  See Sub-Section 6.1.10 to see how this file is used.

 

TRUNC.C

"Trunc.c" truncates, or chops, a floating-point value to SIG significant digits.  None of the algorithms use this function, but it can easily replace "round.c."

 

 

3.4 Documentation Files

 

Previous versions of "Numerical Analysis Algorithms in C" consisted of only two document files: "readme.doc" and "math.doc."  With version 4.2, these documents have been consolidated and greatly expanded into this User's Manual ("usersman.doc").  Three document files are included as listed below.

 

README.DOC

"Readme.doc" gives a list of all the algorithms as well as an order form.  This information can also be found inside the User's Manual.

 

REVHIST.DOC

"Revhist.doc" gives a detailed list of all changes made to each version of "Numerical Analysis Algorithms in C".  It lists the additions, corrections, and changes made to each algorithm, to the supporting files, and to the documentation.

 

USERSMAN.DOC

"Usersman.doc" is this User's Manual in DOS text format.  This format is readable by all text editors and word processors.  It can be read using MS-DOS's "type" command or the "list.com" utility included with the diskettes.

 

 

3.5 Utility Files

 

041EE.C

"041ee.c" is an example of how to integrate the equation evaluator routines into an algorithm.

 

041FUN.C

"041fun.c" is an example of Algorithm 4.1 turned into a stand-alone function.

 

CONVERT.C

"Convert.c" is the C source code for a utility which translates text files into standard seven-bit ASCII files.  It is useful before placing these algorithms on non-MS-DOS computers, such as UNIX and VAX computers.  See Section 7.1 - "Convert.c - Converting Files from Extended ASCII to Standard ASCII."

 

CONVERT.EXE

"Convert.exe" is the MS-DOS executable of "convert.c."

 

LISTALL

"Listall" is a text file listing all source code files on the root directory of the distribution disks.  It can be used with "convert.exe" to convert all the programs at once.

 

LISTOUT

"Listout" is a text file listing all output files in the OUT sub-directory of the distribution disks.  It can be used with "convert.exe" to convert all of the output files at once.

 

LIST.COM

"List.com" is an MS-DOS program which acts as a better "TYPE" command.  It uses the arrow keys and other editing keys to view text files.  "List.com" does not allow you to edit files, just view them.  It is a public domain program.  See Section 7.2 - "List.com - A better TYPE Command" for instructions on how to use it.

 

 

3.6 Batch, Script and Command Files

 

Three commands text files are included to simplify the task of compiling and running the algorithms on different computer systems.

 

CC.BAT

"Cc.bat" is an MS-DOS batch file used for compiling, running and viewing a Microsoft C 5.0 program.  It can be easily altered to allow for linking to "naautil.c" and "eqeval.c" object files, speeding up the compile time.  It can also be altered to increase the stack size of a program.

 

CCC

"Ccc" is a UNIX script file used for compiling, running, and viewing a C program.  It can be easily altered to allow for linking to "naautil.c" object code, speeding up the compile time.

 

VAXCC.COM

"Vaxcc.com" is a VAX/VMS command file used for compiling and linking a mathematical VAX C program.  It can be easily altered to allow for linking to "naautil.c" object code, speeding up the compile time.


3.7 File Structure Chart

 

The chart below describes how the files are organized on the distribution diskettes.

 

                                 / (root)

                                    *

      +))))))))0))))))))0))))))))0))2)))))0))))))))0)))))))),

      *        *        *        *        *        *        *                *.C     *.DOC     UTIL    LANGS     IN       OUT      EXE                                  *        *        *        *        *

                        *        *        *        *        *

                       *.*       *       *.IN    *.OUT    *.EXE

                                 *                      (OPTIONAL)

                                 *                                 

   +)))))))))))0)))))))))))0)))))2)))))0)))))))))))0))))))))))),  

   *           *           *           *           *           *

  ADA        BASIC         C          CPP       FORTRAN      PASCAL 

   *           *           *           *           *           *

SIMPSON.ADA SIMPSON.BAS SIMPSON.C  SIMPSON.CPP SIMPSON.FOR SIMPSON.PAS

NAAUTIL.ADA SIMPSON.IN  SIMPSON.H  SIMPSON.HPP SIMPSON.IN  NAAUTIL.INC

SIMPSON.IN  SIMPSON.OUT SIMPSON.IN SIMPSON.IN  SIMPSON.OUT NAAMATH.INC

SIMPSON.OUT            SIMPSON.OUT SIMPSON.OUT             SIMPSON.IN

                                                           SIMPSON.OUT

 

 

3.8 File Name Translation Table from 3rd to 4th Edition

 

This translation table correlates the third edition text algorithms with the fourth edition text algorithms.  The B and C extensions indicate algorithms that were changed or replaced from the third edition and retained with the fourth edition algorithms.           

                                                       

Edition * Edition       Edition * Edition       Edition * Edition

  3rd   *   4th           3rd   *   4th           3rd   *   4th

))))))))3)))))))))      ))))))))3)))))))))      ))))))))3)))))))))

  2.1   *   2.1           5.3   *   5.3           8.6   *   9.2

  2.2   *   2.2           5.4   *   5.4           8.7   *   9.3

  2.3   *   2.3           5.5   *   5.5           8.8   *   9.4

  2.4   *   2.4           5.6   *   5.6           8.9   *   9.5

  2.5   *   2.5           5.7   *   5.7           8.10  *   9.6B

  2.6   *   2.6           5.8   *   5.8           9.1   *  10.1

  2.7   *   2.7           6.1   *   6.1           9.2   *  10.2

  3.1   *   3.1           6.2   *   6.2           9.3   *  10.3

  3.2   *   3.2           6.3   *   6.3          10.1   *  11.1

  3.3   *   3.3           6.4   *   6.4          10.2   *  11.2

  3.4   *   3.4           6.5   *   6.4C         10.3   *  11.3

  3.5   *   3.5           6.6   *   6.6          10.4   *  11.4

  4.1   *   4.1           6.7   *   6.7          10.5   *  11.5

  4.2   *   4.2           8.1   *   7.1          10.6   *  11.6

  4.3   *   4.3           8.1   *   7.1          11.1   *  12.1

  4.4   *   4.4           8.2   *   7.2          11.2   *  12.2

  5.1   *   5.1           8.3   *   7.3          11.3   *  12.3

  5.2   *   5.2           8.4   *   7.4          11.4   *  12.4

                          8.5   *   9.1          11.5   *  12.5

                                                       

 

3.9 4th Edition Differences                             

                                                        

In the fourth edition's PREFACE, pages vii-viii list the "CHANGES IN THE FOURTH EDITION".  The specifics of these changes are listed below.

 

Renamed Algorithms:                      4.1, 4.4, 7.1, 7.2, 9.2, 10.1, 11.2

New to 4th Edition:                           4.5, 6.5, 9.6

Modified in 4th Edition:            9.5B

Discontinued in 4th Edition:              6.4C, 9.6B



 

 

 

4.  Step-By-Step Examples on Various Computers

 

 

This chapter gives four step-by-step examples on several different computer systems.  The example will use Algorithm 4.1 - Composite Simpson's Rule for Integration ("041.c") and will compute the integral of f(x) = 2*cos(x) from 1 to 2 using 20 intervals.

 

Eight steps are typical every time an algorithm is used.  These steps are:

 

          Step #1 - Change to Correct Directory      (operating system)

          Step #2 - Retrieve Algorithm                      (editor)

          Step #3 - Edit Algorithm                                       (editor)

          Step #4 - Save Modifications                     (editor)

          Step #5 - Compile Algorithm                      (compiler)

          Step #6 - Run Program                              (operating system)

          Step #7 - View Output                                (operating system)

          Step #8 - Print Output                                (operating system)

 

For two-thirds of the algorithms, Steps 2-4 are unnecessary and Step 5 needs to be done only once.  These files are marked with an asterisk ('*') in the table in Section 3.1.

 

The examples below will cover these eight steps on four different computer systems:  MS-DOS PCs, UNIX, Macintoshes, and VAXes.  Before following any of these examples, first check the need list below and configure your "naautil.c" file.

 

 

4.1 Need List

 

For this example the files "naautil.c" and "041.c" are needed.  "Naautil.c" and "041.c" are listed in Appendices A and B to be conveniently referred to during this example.  A simple text editor and a C compiler are also required.  The C compiler should be ANSI compatible if at all possible.  This will save you from possible incompatibility problems.

 

It is recommended that you try this example out on your computer system as you read this section.  Be sure to modify only COPIES of the original algorithms so the algorithms can be used over and over again without problems.

 

 

4.2 Customizing Naautil.c

 

The first decisions to be made are what options and flags you would like to use or set inside the "naautil.c" file.  These flags are usually set only once.  An explanation of each flag is given below.

 

ANSI:

If your compiler supports the ANSI C standard, then set ANSI to TRUE.  Set ANSI to FALSE only if the program will not compile with it set to TRUE.  This flag mostly effects function prototype styles.

 

ANSI_FUNCT:

Set this flag to TRUE to use the ANSI style for declaring functions over the K&R style.  This flag must be set to TRUE if using THINK C 4.0 on a Macintosh.

 

FILE_SAVE:

If you would like to save the output to a file, then set FILE_SAVE to TRUE.  The output is still printed to the screen as you run the program.  Set it to FALSE if you do not want to save the output to a file.

 

TITLE_PROMPT:

If you would like to be prompted for an optional title at the start of each program, then set TITLE_PROMPT to TRUE.  This is useful when the output is to be handed in as homework, allowing the user's name or the problem number to be entered.  No title is printed to the output file if the [ENTER] key is hit by itself.  Set it to FALSE if you do not want to be bothered with entering a title every time you run an algorithm.

 

EQ_EVAL:

Several of the algorithms require a single function to be evaluated.  Set EQ_EVAL to TRUE if you wish to enter the function during run-time instead of at compile time.  A couple of simple modifications MUST be made to your algorithm BEFORE this option will be effective.  See Chapter 8 - "The Equation Evaluator Routines" for instructions on using this option.

 

NAAUTIL_OBJ:

This option is useful for users who wish to speed up the compilation process.  See Section 6.6 - "Using Naautil.c as Object Code" for more details.

 

These examples assume the following default settings:

 

        FLAG           SETTING

        ANSI           TRUE

        ANSI_FUNCT     FALSE

        FILE_SAVE      TRUE

        TITLE_PROMPT   TRUE

        EQ_EVAL        FALSE   (Is set to TRUE in "041ee.c")

        NAAUTIL_OBJ    FALSE

 

The ANSI, ANSI_FUNCT and OLD_UNIX_OS flags may need to be changed if your compiler varies from the ANSI standard.  See Section 6.5 - "Explanation of the Naautil.c File" for a more thorough explanation of the "naautil.c" flags.

 

 

4.3 Example Using MS-DOS, Microsoft C and the P-Edit Editor

 

This example uses the following software:

 

          Operating System:        MS-DOS on an IBM PC

          Compiler:                       Microsoft C 5.0

          Editor:                            WordPerfect's P-Edit Editor

 

No special "naautil.c" flags need to be set.

 

This example assumes the files were installed onto the "C" drive in the "\NAA42" sub-directory.  The DOS prompt will be represented by "C:\NAA42> ".

 

 

Step #1 - Change to Correct Directory

 

Assuming the "Numerical Analysis Algorithm in C" files are located in the "\NAA42" sub-directory of the "C" drive, go there by typing:

 

          C:\> CD \NAA42                                - changes directories

          C:\NAA42> DIR /P                           - shows directory's contents

 

 

Step #2 - Retrieve Algorithm

 

Invoke your text editor and retrieve the algorithm file:

 

          C:\NAA42> PE 041.C

 

The file "041.c" is now loaded and is ready for editing.  A text editor is preferred over a word processor.  If you plan to use a word processor as your editor, be sure to retrieve and save all files as text-only files.

 

 

Step #3 - Edit Algorithm

 

You must now modify the function f(x).  F(x) is listed twice - once as text and once as the actual function call.  All functions are defined at the top of each program.  To quickly find where modifications are necessary, search for the '$' character.  This character is used exclusively for locating lines of code that need updating in all "Numerical Analysis Algorithms in C" files.

 

Search for the first '$':

 

          [F2] $ [F2]                              - search

 

The first '$' should be found at line 22 of "041.c."

 

Change line 22 from:     char *eq_text_f = "f(x) = sin(x)";

to:                char *eq_text_f = "f(x) = 2*cos(x)";

 

This string of text will be printed as output exactly as it appears inside the quotations when the program is run.

 

Now search for the second '$':

 

          [F2] $ [F2]                              - search

 

The second '$' should find the function itself on line 31 of "041.c."

Change line 31 from:     return (sin(x));

to:                return (2.0 * cos(x));

 

 

Step #4 - Save Modifications

 

Now save the file "041.c" with the above changes and exit the editor:

 

          [F7] Y [ENTER] Y Y             - save and exit

 

 

Step #5 - Compile Algorithm

 

Now compile and link "041.c" into the executable file "041.exe."  At the prompt type:

 

          C:\NAA42> CL 041.C

 

The batch file "cc.bat" can also be used in place of the "CL" command.  See Sub-Section 7.3.1 on using "cc.bat."  If the program requires a larger stack than the default size, using "CL 041.C /link /ST:4096" will increase the stack from 2K bytes to 4K bytes in Microsoft C 5.0.

 

 

Step #6 - Run Program

 

To run "041.exe", at the DOS prompt type:

 

          C:\NAA42> 041

 

The ".exe" extension can be left off.  Answer the prompts with the predetermined inputs.  The screen should look something like this:

 

    64444444444444444444444444444444444444444444444444444444444447

    5 ‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑ 5

    5          "Numerical Analysis Algorithms in C" v4.2         5

    5 ‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑ 5

    5                                                            5

    5 Enter an optional title [ie ‑ Set 2.1,  Problem 2 a) ].    5

    5 ‑‑‑‑> User's Manual Example                                5

    5                                                            5

    5 Composite Simpson's Rule ‑ Algorithm 4.1                   5

    5                                                            5

    5 f(x) = 2*cos(x)                                            5

    5                                                            5

    5 Enter endpoint a: 1                                        5

    5 Enter endpoint b: 2                                        5

    5 Enter number of intervals on [a,b], n: 20                  5

    5 Interval number h = 0.05                                   5

    5                                                            5

    5      !2                                                    5

    5 XI = *  f(x) dx = 0.13565288875                            5

    5      "1                                                    5

    5                                                            5

    5 Required 21 functional evaluations.                        5

    5                                                            5

    5 Output saved into file "041.out".                          5

    94444444444444444444444444444444444444444444444444444444444448

 

As indicated by the output, a file named "041.out" is created which contains the output results in a ready-to-print format.

 

 

Step #7 - View Output

 

To view the contents of the output file "041.out", use either the DOS "type" command, the "Numerical Analysis Algorithms in C" utility "list.com", or your text editor.  See Section 7.2 for instructions on the usage of the "list.com" utility.

 

          C:\NAA42> TYPE 041.OUT                                - Using DOS's "type"

or

          C:\NAA42> UTIL\LIST 041.OUT                             - Using "list.com"

 

If the file's contents are accurate, then you are ready to print out a copy to be turned in as homework.

 

 

Step #8 - Print Output

 

To print out the output file from DOS, type:

 

          C:\NAA42> PRINT 041.OUT

 

This step can also be done from within most text editors.  WARNING: Be careful not to print the executable file "041.exe".  It will waste reams of paper.

 

 

4.4 Example Using UNIX, cc and the vi Editor

 

This example uses the following software:

 

          Operating System:        UNIX

          Compiler:                       cc

          Editor:                            vi

 

You may need to set the OLD_UNIX_OS flag to TRUE if your C compiler requires the include file <varargs.h> instead of <stdarg.h> for variable length argument lists.  See your system's "/usr/include" sub-directory to determine which include file will be used.

 

The percent ('%') character will be used to represent the UNIX shell prompt.

 

 

Step #1 - Change to Correct Directory

 

Assuming the "Numerical Analysis Algorithm in C" files are located in the "naa42" sub-directory, go there by typing:

 

          % cd naa42                                          - changes directories

          % pwd                                                       - shows current directory

          % ls -alF                                             - shows directory's contents

 

 

Step #2 - Retrieve Algorithm

 

Invoke the vi editor and retrieve the algorithm file:

 

          % vi 041.c

 

The file "041.c" is now loaded and is ready for editing.

 

 

Step #3 - Edit Algorithm

 

You must now modify the function f(x).  F(x) is listed twice - once as text and once as the actual function call.  All functions are defined at the top of each program.  To quickly find where modifications are necessary, search for the '$' character.  This character is used exclusively for locating lines of code that need updating in all "Numerical Analysis Algorithms in C" files.

 

Search for the first '$':

 

          /$                                           - search

 

The first '$' should be found at line 22 of "041.c."

 

Change line 22 from:     char *eq_text_f = "f(x) = sin(x)";

to:                char *eq_text_f = "f(x) = 2*cos(x)";

 

This string of text will be printed as output exactly as it appears inside the quotations when the program is run.

 

Now search for the second '$':

 

          n                                              - search (next occurrence)

 

The second '$' should find the function itself on line 31 of "041.c."

 

Change line 31 from:     return (sin(x));

to:                return (2.0 * cos(x));

 

Here are a few vi editing commands you should know for future reference:

 

          i                   Enters insert mode  (Exit this mode using [ESC])

          R                 Enters typeover mode  (Exit this mode using [ESC])

          r                  Replace character

          w                 Moves forward one word

          b                 Moves backward one word

          x                  Deletes a character

          dw               Deletes a word

          dd               Deletes a line

          cw               Changes a word  (follow text by an [ESC] key)

          :#                Go to line number #

          :w                Saves (writes) editor contents to a file

          :q                Quits (exits) the editor

          ZZ               Exits the editor saving all changes  (Same as ":wq")

          [ESC]          Exits insert, typeover, and other editing modes

          /string         Searches forward for string

          ?string        Searches backwards for string

          n                 Continue search for string

          Arrow keys, ^g, ^h, ^j, ^k, or [SPACE] move the cursor

 

 

Step #4 - Save Modifications

 

Now save the file "041.c" with the above changes and exit the editor:

 

          :wq                                         - write and quit

or

          ZZ                                           - save and exit (faster to type than ":wq")

 

 

Step #5 - Compile Algorithm

 

Now compile and link "041.c" into the executable file "041".  At the shell prompt type:

 

          % cc -o 041 041.c -lm

 

"Cc" invokes the C compiler, "-o 041" (NOT ‑0) names the executable program, "041.c" is the source code file name, and "-lm" links with the math library.  Without the "-o 041" the program would be given the default name of "a.out".  Without the "-lm" the program would give incorrect floating-point results.

 

The script file "ccc" can also be used in place of the "cc" command.  See Sub-Section 7.3.2 on using "ccc".  It will do the compiling, running, and will list the output for you.

 

 

Step #6 - Run Program

 

To run "041", at the shell prompt type:

 

          % 041

 

Answer the prompts with the predetermined inputs.  The screen should look something like this:

 

    64444444444444444444444444444444444444444444444444444444444447

    5 ‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑ 5

    5          "Numerical Analysis Algorithms in C" v4.2         5

    5 ‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑ 5

    5                                                            5

    5 Enter an optional title [ie ‑ Set 2.1,  Problem 2 a) ].    5

    5 ‑‑‑‑> User's Manual Example                                5

    5                                                            5

    5 Composite Simpson's Rule ‑ Algorithm 4.1                   5

    5                                                            5

    5 f(x) = 2*cos(x)                                            5

    5                                                            5

    5 Enter endpoint a: 1                                        5

    5 Enter endpoint b: 2                                        5

    5 Enter number of intervals on [a,b], n: 20                  5

    5 Interval number h = 0.05                                   5

    5                                                            5

    5      [2                                                    5

    5 XI = |  f(x) dx = 0.13565288875                            5

    5      ]1                                                    5

    5                                                            5

    5 Required 21 functional evaluations.                        5

    5                                                            5

    5 Output saved into file "041.out".                          5

    94444444444444444444444444444444444444444444444444444444444448

 

As indicated by the output, a file named "041.out" is created which contains the output results in a ready-to-print format.

 

 

Step #7 - View Output

 

To view the contents of the output file "041.out", use the UNIX "more" command.

 

          % more 041.out

 

If the file's contents are accurate, then you are ready to print out a copy to be turned in as homework.

 

 

Step #8 - Print Output

 

To print out the output file from the UNIX shell prompt, type:

 

          % lp 041.out

 

"Lp" prints the file "041.out" to the line printer.  WARNING: Never try to print the executable file "041*" (denoted with an '*' when listed with "% ls -F").  It will waste reams of paper.

 

 

4.5 Example Using a Macintosh and THINK C

This example uses the following software:

 

          Operating System:        Finder or MultiFinder on a Macintosh

          Compiler:                       THINK C 4.0 by Symantec

          Editor:                            THINK C editor

 

You will need to set the ANSI_FUNCT flag in "naautil.c" to TRUE to compile and use functions using variable length argument lists, such as "printf2(...)" and "eval_eq()".  It simply enforces the newer ANSI style function declarations over the older K&R style (see Section 9.1 for an example).

 

The following example was derived from Chapter 3 - "Tutorial: Hello World" in the THINK C User's Manual.  It replaces the "Hello Folder" with "041 Folder.B", "hello.c" with "041.c", and uses the ANSI library.

 

 

Step #1 - Create a Project

 

The first thing you need to do is create a folder called "041 Folder.B" in the "Development" folder.  Do this before you start THINK C.  The "041 Folder.B" folder should contain your source files ("041.c"), "naautil.c" and other supporting ".c" files such as "eqeval.c".  It is good programming practice, though not necessary, to name your project folders with a ".B" extension.  (To make a B, type Option p.)

 

When you've created "041 Folder.B", open the THINK C Folder (the one that contains the THINK C application) and double click on the THINK C icon.

 

You'll see a dialogue box that asks you to open a project.  Since you are creating a new project, click on the New button.  You'll see another dialogue box, one that lets you create projects.

 

Move back to the "041 Folder.B" folder you just created.  It is very important that you move to this folder.  Name the project "041 project", and click on the Create button.  THINK C creates a new project document on disk and displays a project window.

 

 

Step #2 - Retrieve Algorithm

 

To open the algorithm text file, choose the Open... command in the File menu.  Select the file "041.c" from the menu.

 

 

Step #3 - Edit Algorithm

 

You must now modify the function f(x).  F(x) is listed twice - once as text and once as the actual function call.  All functions are defined at the top of each program.  To quickly find where modifications are necessary, search for the '$' character.  This character is used exclusively for locating lines of code that need updating in all "Numerical Analysis Algorithms in C" files.

 

To search for the first '$' character, choose the Find... command in the Search menu.  Type a '$' character in the Search for: field and click the Find button.  It should be found at line 22 of "041.c."

 

Change line 22 from:     char *eq_text_f = "f(x) = sin(x)";

to:                char *eq_text_f = "f(x) = 2*cos(x)";

 

This string of text will be printed as output exactly as it appears inside the quotations when the program is run.

 

Now search for the second '$' by choosing the Find Again command in the Search menu.  The second '$' should find the function itself on line 31 of "041.c."

 

Change line 31 from:     return (sin(x));

to:                return (2.0 * cos(x));

 

You may want to read Chapter 8 - "The Editor" in your THINK C User's Manual for more information about the THINK C text editor.

 

 

Step #4 - Save Modifications

 

When you have finished modifying the program, select Save As... from the File menu to save it.  You will get a dialogue box in which you should enter the name of the file "041.c", and click on the Save button.  THINK C will only compile files that end with ".c" or ".C".

 

 

Step #5 - Compile Algorithm

 

Now compile "041.c" into the executable named "041".  To do this, select Compile from the Source menu.  THINK C displays a dialogue box that shows how many lines have been compiled.  See your THINK C User's Manual if you can not resolve any compilation errors.

 

Next, you need to add the "ANSI" library to your project.  This library contains all the standard C library routines such as printf().  To add the "ANSI" library, choose Add... from the Source menu.

 

When you get the standard file dialogue box, open the folder called "C Libraries."  This folder contains all the libraries for ANSI compatibility, including the "ANSI" library.  Select "ANSI", and click on the Add button.  WARNING - Do not select "ANSI-small" or "ANSI-A4" since they do not support floating-point operations.  If you have a math coprocessor (MC68881), substitute "ANSI" with "ANSI-881".  This will measurably speed up each algorithm's execution time.

 

THINK C adds the name "ANSI" to the project window and then puts up the standard file dialogue box again.  The second time around just click on the Cancel box.  THINK C will load the library automatically when you run the project.

 

IMPORTANT:  You may need to place "ANSI" into its own segment by dragging "ANSI" below the dotted line and releasing it.  A line indicates that the code is separated into different segments.  This may be necessary due to an object code size limitation of 32K bytes per segment.

 

 

Step #6 - Run Program

 

Everything is all set to run the project.  The source file is in the project window along with the libraries you will be using.  Now select Run from the Project menu.

THINK C notices that the library needs to be loaded, so it puts up a dialogue box asking you if you want to bring the project up to date.  Click on the Yes button.  THINK C goes to disk to load the code for the "ANSI" library.  The executable "041" is now being created.

 

Since all "Numerical Analysis Algorithms in C" programs call the printf() function, all output will go to a window called "console".  The console window emulates a generic terminal screen.

 

The program will now prompt you for inputs.  Answer the prompts with the predetermined inputs.  The console screen should look something like this:

 

    64444444444444444444444444444444444444444444444444444444444447

    5////////////////////////// console ///////////////////////G/5

    :444444444444444444444444444444444444444444444444444444444444<

    5 ‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑ 5

    5          "Numerical Analysis Algorithms in C" v4.2         5

    5 ‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑ 5

    5                                                            5

    5 Enter an optional title [ie ‑ Set 2.1,  Problem 2 a) ].    5

    5 ‑‑‑‑> User's Manual Example                                5

    5                                                            5

    5 Composite Simpson's Rule ‑ Algorithm 4.1                   5

    5                                                            5

    5 f(x) = 2*cos(x)                                            5

    5                                                            5

    5 Enter endpoint a: 1                                        5

    5 Enter endpoint b: 2                                        5

    5 Enter number of intervals on [a,b], n: 20                  5

    5 Interval number h = 0.05                                   5

    5                                                            5

    5      [2                                                    5

    5 XI = |  f(x) dx = 0.13565288875                            5

    5      ]1                                                    5

    5                                                            5

    5 Required 21 functional evaluations.                        5

    5                                                            5

    5 Output saved into file "041.out".                          5

    94444444444444444444444444444444444444444444444444444444444448

 

To exit the program, press the Return key or choose Quit from the File menu.

 

As indicated by the output, a text file named "041.out" is created which contains the output results in a ready-to-print format.

 

 

Step #7 - View Output

 

To view the contents of the output file, use the Open... command in the File menu and select "041.out."

 

If the file's contents are accurate, then you are ready to print out a copy to be turned in as homework.

 

Step #8 - Print Output

 

To print out the output file, use the Print... command in the File menu.  Make sure the output file is in the frontmost edit window.  You'll see the standard print dialogue for either the ImageWriter or LaserWriter.

 

To end this example session, select Close All in the Windows menu to close all open files.  If a file has not been saved, the editor will ask you if you want it saved.

 

 

Using SANE

 

As you use these algorithms, you may find it beneficial to use certain utility functions from the Standard Apple Numerics Environment (SANE).  The SANE library uses 80-bit values and is not intended for projects that have the MC68881 Code Generation option checked.

 

The eight functions below are common to both the SANE and ANSI libraries:

 

        atan()         exp()          log()          sqrt()

    cos()       fabs()      sin()       tan()

 

To use SANE versions, #include the file "SANE.h" before the file "math.h" inside "naautil.c."  Similarly, to use the ANSI versions, #include the file "math.h" before the file "SANE.h" in "naautil.c."  For more information on SANE, read "Apple Numerics Manual, Second Edition" (Addison-Wesley).

 

 

4.6 Example Using VAX/VMS, CC and the EDIT/EDT Editor

 

This example uses the following software:

 

          Operating System:        VAX/VMS (really DCL)

          Compiler:                       VAX C v3.2 from DEC (CC)

          Editor:                            EDIT/EDT or EVE

 

The dollar ('$') character will be used to represent the VMS command prompt.

 

 

Step #1 - Change to Correct Directory

 

Assuming the "Numerical Analysis Algorithm in C" files are located in the "NAA42" sub-directory, go there by typing:

 

          $ SET DEFAULT [.NAA42]            - changes directories

          $ SHOW DEFAULT                                - shows current directory

          $ DIR/SIZE/DATE                              - shows directory's contents

 

 

Step #2 - Retrieve Algorithm

 

Invoke the EDIT/EDT editor and retrieve the algorithm file:

 

          $ EDIT/EDT 041.C

 

The file "041.c" is now loaded and is ready for editing.  The first line of the file is printed to the screen.  An asterisk will follow which indicates that you are in EDT line editing mode.  It should look similar this:

 

          $ EDIT/EDT 041.C

              1       /*****************************************************

****************

          *

 

 

Step #3 - Edit Algorithm

 

Now type "C" or "SET MODE CHANGE" followed by [ENTER] to leave line editing mode and enter full screen mode where you can use the EDT function keypad.

 

          * C [ENTER]

 

You must now modify the function f(x).  F(x) is listed twice - once as text and once as the actual function call.  All functions are defined at the top of each program.  To quickly find where modifications are necessary, search for the '$' character.  This character is used exclusively for locating lines of code that need updating in all "Numerical Analysis Algorithms in C" files.

 

Search for the first '$' by entering:

 

          [4] [PF1] [PF3] $

 

The first '$' should be found on line 22 of "041.c."

 

Change line 22 from:     char *eq_text_f = "f(x) = sin(x)";

to:                char *eq_text_f = "f(x) = 2*cos(x)";

 

This string of text will be printed as output exactly as it appears inside the quotations when the program is run.

 

Now search for the second '$' by entering:

 

          [4] [PF1] [PF3] $

 

The second '$' should find the function itself on line 31 of "041.c."

 

Change line 31 from:     return (sin(x));

to:                return (2.0 * cos(x));

 

Here are a few EDIT/EDT editing commands you should know:  (^ = [CONTROL])

 

          [PF2]           Help

          [PF1][0]       Opens blank line after current line

          [,]                Replace character

          [4][1]           Moves forward one word

          [5][1]           Moves backward one word

          [.]                Deletes a character

          [-]                Deletes a word  (Must be followed by the [ESC] key)

          [PF4]           Deletes a line

          [-]                Changes a word  (Must be followed by the [ESC] key)

          [PF1][7]T#  Moves to line number #

          ^Z EXIT       Quits the editor and saves any changes

          ^Z QUIT       Quits the editor without saving changes

          [ESC]                   Terminate input mode

          ^Z                Exits full-screen mode and returns to line mode with *

          [4][PF1][PF3]string                  Searches forward for string

          [5][PF1][PF3]string                  Searches backwards for string

          [PF1][7] EXIT [ENTER]  Exits editor saving any changes

          Arrow keys, ^g, ^h, ^j, ^k, or [SPACE] move the cursor

 

 

Step #4 - Save Modifications

 

Now save the file "041.c" with the above changes and exit the editor:

 

          ^Z                                  - returns to line editing mode and the * prompt

          * EXIT                        - save and exit

 

 

Step #5 - Compile Algorithm

 

The VAX C compiler needs to know which libraries to link to.  Two libraries will be used which will allow floating-point operations.  Define them once as follows:

 

          $ DEFINE LNK$LIBRARY    SYS$LIBRARY:VAXCRTLG

          $ DEFINE LNK$LIBRARY_1  SYS$LIBRARY:VAXCRTL

 

See "HELP CC Link_libraries" to make sure the defines above are correct for your VAX as well (/G_FLOAT without Curses).

 

Now compile and link "041.c" into the executable file "041.exe".  At the VAX prompt type:

 

          $ CC /G_FLOAT 041.C

          $ LINK 041, LNK$LIBRARY/LIB, LNK$LIBRARY_1/LIB

 

"Cc" compiles "041.c" into "041.obj" object code.  "Link" names the executable "041.exe" after linking it to the appropriate libraries.  For machine specific information on the "link" command, use the on-line help by typing "HELP CC LINK" and "HELP LINK."

 

The command file "vaxcc.com" can also be used in place of the "cc" and "link" commands.  See Sub-Section 7.3.3 on using "vaxcc.com".  It will do the compiling and linking in one simple step, assuming the link libraries are correct.  Using it is as easy as typing:

 

          $ @VAXCC.COM 041                  - replaces Step #5 entirely

 

 

Step #6 - Run Program

 

To run "041.exe", at the VAX prompt type:

 

          $ RUN 041

 

Answer the prompts with the predetermined inputs.  The screen should look something like this:

 

    64444444444444444444444444444444444444444444444444444444444447

    5 ‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑ 5

    5          "Numerical Analysis Algorithms in C" v4.2         5

    5 ‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑ 5

    5                                                            5

    5 Enter an optional title [ie ‑ Set 2.1,  Problem 2 a) ].    5

    5 ‑‑‑‑> User's Manual Example                                5

    5 Composite Simpson's Rule ‑ Algorithm 4.1                   5

    5                                                            5

    5 f(x) = 2*cos(x)                                            5

    5                                                            5

    5 Enter endpoint a: 1                                        5

    5 Enter endpoint b: 2                                        5

    5 Enter number of intervals on [a,b], n: 20                  5

    5 Interval number h = 5.000000e-02                           5

    5                                                            5

    5      [2                                                    5

    5 XI = |  f(x) dx = 0.13565288875                            5

    5      ]1                                                    5

    5                                                            5

    5 Required 21 functional evaluations.                        5

    5                                                            5

    5 Output saved into file "041.out".                          5

    94444444444444444444444444444444444444444444444444444444444448

 

As indicated by the output, a file named "041.out" is created which contains the output results in a ready-to-print format.

 

 

Step #7 - View Output

 

To view the contents of the output file "041.out", use the "TYPE" command.

 

          $ TYPE/PAGE 041.OUT

 

If the file's contents are accurate, then you are ready to print out a copy to be turned in as homework.

 

 

Step #8 - Print Output

To print out the output file from the VMS prompt, type:

 

          $ PRINT 041.OUT

 

WARNING: Never try printing the executable file "041.exe."  It will waste reams of paper.



 

 

 

5.  For Those New to C

 

 

This chapter will introduce you to the C programming language and some of its basic functions and features.  if you are new to C, it will be to your advantage to take a few minutes to read through this chapter before you move on.  If you are already familiar with C, you may want to glance through this chapter to remind you of the math library functions found in <math.h>.

 

The C language has been around since 1978.  Its popularity continues to grow especially among universities and industry.  C is usually learned as a second language after learning Pascal or FORTRAN.  This chapter is intended to give unexperienced programmers a push in the right direction.

 

The easiest way to learn C is by example.  This chapter also lists the preferred reference books, the mathematical operators and functions, and compares C with other popular programming languages -- along with examples.

 

If you do not own a C compiler and you have access to an IBM PC computer, and you do not want to pay much to get one (student mode), there are some low cost compilers on the market that you may wish to investigate.  One such compiler is "Power C".  This ANSI compatible C compiler lists for only $19.95.  To order, call 1-800-333-0330, or write to: MIX Software, 1132 Commerce Dr., Richardson, TX 75081, (214) 783-6001.  Turbo C and Microsoft C seem to be among the most popular DOS C compilers on the market.

 

The definitive book on the C language is "The C Programming Language", Second Edition, by Brian W. Kernighan and Dennis M. Ritchie (Cost: $28.00).  If you are using an older C compiler (pre-1987), you may find the first edition more useful.  This 272 page book was written by the creators of C at AT&T Bell Laboratories.  All other books on C are derivatives of this book.

 

The syntax of older C compilers follows the first edition of "The C Programming Language."  This pre-standard is often referred to as K&R style, named after its authors, Kernighan and Ritchie.  The second edition was revised to conform to the ANSI standard.

 

 

5.1 Mathematical Operators

 

The following operators are used to write mathematical equations in C.  These operators are built-in to the C language.  For more detailed descriptions, see your C compiler's documentation.

 

Operator              Description

*                          Multiplication.  Not to be confused with pointers.

                             Example:  a = b * c;

 

/                          Division.  Chops to nearest integer if using integer types.  For instance,  11 / 4 = 2 since the remainder of 3 is discarded.  11.0 / 4.0 = 2.75.

                   Example:  a = b / c;

 

%                          Remainder.  Also called the modulus operator.  Use fmod() and/or modf() for floats and doubles.  For instance, 11 % 4 = 3 since the quotient of 2 is discarded.

                   Example:  a = b % c;

 

+                          Addition.

                             Example:  a = b + c;

 

                          Subtraction and arithmetic negation.

                             Example:  a = b ‑ c;  and  a = ‑b;

 

++                        Increment.  For instance, i++; is shorthand for i = i + 1;

                             Example:  i++;  (post‑increment)  and  ++i;  (pre‑increment)

 

‑‑                        Decrement.  For instance, i‑‑; is shorthand for i = i ‑ 1;

                             Example:  i‑‑;  (post‑decrement)  and  ‑‑i;  (pre‑decrement)

 

*=                        Multiplication assignment.  For instance, x *= 3.14 + y; is shorthand for x = x * (3.14 + y);

 

/=                        Division assignment.  For instance, x /= 3.14 + y; is shorthand for x = x / (3.14 + y);

 

%=                        Remainder assignment.  Integers only.  For instance, a %= 314 + b; is shorthand for a = a % (314 + b);

 

+=                        Addition assignment.  For instance, x += 3.14 + y; is shorthand for x = x + (3.14 + y);

 

‑=                        Subtraction assignment.  For instance, x ‑= 3.14 + y; is shorthand for x = x ‑ (3.14 + y);

 

 

5.2 Mathematical Functions

 

The following functions are useful when writing mathematical equations in C.  These functions are not part of the C language proper, but are part of the standard library, an environment that supports standard C.  For more detailed descriptions of these libraries, see your C compiler's documentation.  Another good place to browse is inside the include files <math.h> and <stdlib.h>.  These two include files provide the function declarations for most of the below functions.

 

Listed below are the variable types used in the examples.

 

        Type           Variables

        float          w;

        double         x, y, exp;

        int            *expptr, *intptr, n;

        long int       p, q;

        char           *string;

        div_t          num, denom;

        struct complex z;

 

struct complex { double r,i; } z;  /* Real and imaginary components */

 

#include <math.h>               - must be included to use these functions!

#include <stdlib.h>          - must be included to use these functions!

 

 

Function               Description

abs(n)              Returns the absolute value of its integer argument.

 

acos(x)           Returns the arccosine of x in the range 0 to B.  The value of x must be between ‑1 and 1.

 

asin(x)           Returns the arcsine of x in the range ‑B/2 to B/2.  The value of x must be between ‑1 and 1.

 

atan(x)           Returns the arctangent of x in the range ‑B/2 to B/2.

 

atan2(y,x)    Returns the arctangent of y/x in the range ‑B to B.  Unlike atan(), atan2() uses the signs of both x and y to determine the true quadrant of the return value.

 

atof(string)         Converts a character string into a double‑precision floating‑point value.

 

atoi(string)         Converts a character string into an integer value.

 

cabs(z)           Returns the absolute value of a complex number, which must be a structure of type complex (shown above).   Equivalent to sqrt(z.x*z.x + z.y*z.y).  NOT IN ANSI STANDARD.

 

ceil(x)           Returns a double‑precision floating‑point value representing the smallest integer not less than x.  Also called the postage stamp function.

                             Example: ceil(1.05) = 2.0,  ceil(‑1.05) = ‑1.0

 

cos(x)              Returns the cosine of x, where x is in radians.

 

cosh(x)           Returns the hyperbolic cosine of x.

 

div(num,denom)    Computes the quotient and remainder of num/denom.  The results are stored in the int members quot and rem of a structure of type div_t.

 

exp(x)              Returns the exponential function of its floating‑point argument x.  Also called Euler's or the natural number, e . 2.71828182845.

 

fabs(x)           Returns the absolute value of its floating‑point argument x.

 

floor(x)         Returns a double‑precision floating‑point value representing the largest integer not greater than x.  Also called the greatest integer function, [ ].

                             Example: floor(1.05) = 1.0,  floor(‑1.05) = ‑2.0

 

fmod(x,y)      Returns the floating‑point remainder f of x/y such that x = i*y + f, where i is an integer.  f has the same sign as x, and the absolute value of f is less than the absolute value of y.  If y is zero, the result is implementation defined.

frexp(x,expptr) Breaks down the floating‑point value, x, into a mantissa, p, and an exponent, q, such that the absolute value of p is $ 0.5 and < 1.0, and x = p*2^q.  The integer exponent is stored in the location pointed to by expptr.  If x is zero, both parts of the result are zero.

 

hypot(x,y)    Returns the length of the hypotenuse of a right triangle, given the length of the two sides x and y.  Equivalent to:  sqrt(x*x + y*y).  NOT IN ANSI STANDARD.

 

ldexp(x,exp)         Returns x * 2^exp.

 

log(x)              Returns the natural logarithm of x, x > 0.

 

log10(x)         Returns the base‑10 logarithm of x, x > 0.

 

modf(x,intptr)    Breaks down the floating-point value x into fractional and integer parts.  The signed fractional portion of x is returned.  The integer portion is stored as a floating‑point value at intptr.

 

pow(x,y)         Returns x raised to the yth power (x^y).  A domain error occurs if x = 0 and y # 0, or if x # 0 and y is not an integer.

 

rand()              Returns a pseudo‑random integer in the range 0 to RAND_MAX, which is at least 32,767.

 

sin(x)              Returns the sine of x, where x is in radians.

 

sinh(x)           Returns the hyperbolic sine of x.

 

sqrt(x)           Returns the square root of x, x $ 0.

 

srand(seed) Uses seed as the seed for a new sequence of pseudo‑random numbers.  The initial seed is 1.

 

tan(x)              Returns the tangent of x, where x is in radians.

 

tanh(x)           Returns the hyperbolic tangent of x.

 

 

AVAILABLE AS EXTENSIONS ON SOME C COMPILERS (ie ‑ MIPS for an R3000A/R3010):

 

fsin(w)           Sine for floats.  Sin(x) is for doubles.

 

fcos(w)           Cosine for floats.  Cos(x) is for doubles.

 

ftan(w)           Tangent for floats.  Tan(x) is for doubles.

 

fasin(w)         Arcsine for floats.  Asin(x) is for doubles.

 

facos(w)         Arccosine for floats.  Acos(x) is for doubles.

 

fatan(w)         Arctangent for floats.  Atan(x) is for doubles.

 

fsinh(w)         Hyperbolic sine for floats.  Sinh(x) is for doubles.

 

fcosh(w)         Hyperbolic cosine for floats.  Cosh(x) is for doubles.

 

ftanh(w)         Hyperbolic tangent for floats.  Tanh(x) is for doubles.

 

 

5.3 General Language Hints

 

 

Ternary Statements:

 

C has a couple of constructs that may be foreign to users used to FORTRAN 77 or other high level languages.  One of these is the ternary statement:

 

                   a = b ? c : d;

 

which is equivalent to:

 

                   if (b == TRUE)

                             a = c;

                   else

                             a = d;

 

A couple of examples might include:

 

                   max = (a > b) ? (a) : (b);

or

                   printf("%d iteration%s", iter, (iter > 1) ? "s" : "");

                   /* Prints: "1 iteration" and "2 iterations" */

 

 

Defining TRUE and FALSE:

 

Remember, in C "0" is FALSE while anything other than "0" is defined as TRUE.  For example:

 

                   -2      = TRUE

                   -1      = TRUE

                    0       = FALSE

                    1       = TRUE      (default)

                    2       = TRUE

 

Usually, TRUE and FALSE are defined as "#define FALSE 0" and "#define TRUE !FALSE" or "#define TRUE 1".

 

 

Common Equivalents:

 

        SHORT HAND     LONG HAND

        if (expr) ...  if (expr == TRUE) ...

        if (!expr) ... if (expr == FALSE) ...

        i++            i = i + 1

        i--            i = i - 1

        i += 2         i = i + 2

        i -= 2      i = i - 2

 

 

5.4 Language Transition Kit

 

Many numerical analysis students may already be familiar with another programming language other than C.  This section is intended to help those who have learned other languages other than C to transfer their knowledge easily into C.  To accomplish this goal, two large appendices have been compiled.

 

Appendix C contains a set of charts comparing C statements with those of other popular languages.  The tables provided should help in understanding and modifying the equations and code as needed to perform numerical analysis.  These tables show a simple comparison of programming statements most likely to be used in numerical analysis programs.

 

Appendix D contains a set of working examples in six different languages.  These source code examples show how programs look in each of these languages.  These programs do numerical integration using Algorithm 4.1 - Composite Simpson's Rule.  Each program was compiled and run to ensure they were logically and syntactically correct.  The input, output, and include files are also listed for completeness.  These files are included in the LANGS sub-directory on the distribution diskettes.

 

The list below shows the language, compiler and standard used to create the comparison charts and example programs.

 

          LANGUAGE          COMPILER                                                STANDARD                  

1.       Ada                       Meridian Ada 4.1                     ANSI/MIL‑STD‑1815A

2.       BASIC                  Microsoft GW‑BASIC 3.20

3.       C                          Microsoft C 5.0                                 ANSI C

4.       C++                      Borland Turbo C++ 2.0           AT&T C++ v2.0

5.       FORTRAN 77      Microsoft FORTRAN 77 3.3             ANSI FORTRAN 77

6.       Pascal                  Borland Turbo PASCAL 3.01A

 

This language transition kit, comprised of Appendices C and D, account for one-third of this User's Manual.  They are not really a necessary part of the "Numerical Analysis Algorithms in C' package, but they tremendously aid those who are new or "rusty" on their computer programming skills.

 


 

 

 

6.  Helps and Hints

 

 

This chapter contains many of the fine details that can make your use of this software package a pleasant experience.  Read each section as soon as possible to avoid wasting unnecessary time with tasks or problem solving.  The sections below are designed to save you time, improve your confidence in the algorithms, bring your attention to compiler and text errors, and help you customize the programs to best suit your needs.

 

 

6.1 Generally Nice To Know

 

The following sub-sections will give you a better understanding of how to manipulate and customize these algorithms.  They may even save you the trouble of learning any peculiarities of "Numerical Analysis Algorithms in C" the hard way.

 

 

6.1.1        Professor's Favorites, Must Have, Algorithms

 

Six algorithms have been included as requested by several Brigham Young University mathematics professors.  These programs are not included in the text, but serve to enhance it.  In reality, these are the programs that had to be included in order to persuade Brigham Young University to convert from FORTRAN to C.  Each of these programs are named with an "A.c" suffix.  These algorithms are:

 

          028A.c        - Complex Polynomial Solver (CPOLY)

          101A.c        - Steffenson's Method for Systems

          126A.c        - Parabolic Equations With Newton Iteration in 1-D

          127A.c        - Parabolic Equations With Newton Iteration in 2-D

          128A.c        - Elliptic Equations With Newton Iteration in 2-D

          129A.c        - Biharmonic Equation Using Gauss-Jordan Method

 

 

6.1.2        Homework Helper Algorithms

 

Each algorithm not specifically given in the text has a B, C, or D placed before the ".c" extension in its file name.  Roughly a third of all the programs included are modifications to the given text algorithms.  Many of them are requested as homework exercises.  These modifications range from implementing SIG-digit rounding, or adding Richardson's extrapolation, to solving for AX=B after performing matrix factorization.

 

Each program has a comment block at the top of the file.   This comment block also indicates which page of the text and which problem numbers to expect to use these "Homework Helper" algorithms.  This was included to show where these modifications fit into the text.

 

 

6.1.3        Optional Title

 

Each program begins by prompting for a one‑line title.  This title is printed to the output file for your convenience.  If you do not want a title then just enter a [RETURN] or [ENTER] and no title will be used.  To turn off the prompt for an optional title, simply change the TITLE_PROMPT flag to FALSE in the file "naautil.c."

 

 

6.1.4        Optional File Saving

 

Each program has a default output file name associated with it.  This file has the same name as the program being run, but with a ".out" extension.  The default setting in "naautil.c" is to create an output file as a program is run.  To run a program without saving the output to the default output file, just change the FILE_SAVE flag to FALSE in the file "naautil.c."

 

Errors may result if your disk is too full or the disk is write-protected while the FILE_SAVE flag is set to TRUE.

 

 

6.1.5        Finding Functions

 

Many of the algorithms require a function to be evaluated.  These algorithms can be found in chapters 2, 4, 5, 8, 11, and 12.  The functions are printed out to the screen and to the output file.  Each function needs changing in two places, once in the function itself, and once in the comments to be printed out.  Both of these are shown at the top of each program before main().  To aid you in finding these functions, search for the "$" character.  This is the only use of the "$" symbol throughout all the programs.

 

 

6.1.6        Using Default Inputs

 

Several of the programs ask if another input needs to be evaluated.  Make use of the default inputs as shown by just pressing the [ENTER] key.  This will make repetitious loops easier to use. Example:  "Evaluate another value of X? (Y/N) Y"  means just press [ENTER] for Yes.

 

There is no default for entering tolerances (TOL).  When shown one, it is a suggested tolerance, not a default.  Hitting [ENTER] will cause the program to keep waiting (blank stares) until a valid floating-point number is entered.

 

Entering text where numbers are expected or numbers where text is expected will cause the programs to "crash" and usually enter an infinite input loop.  This is characteristic of the scanf() function.  This unfortunate situation can usually be remedied by typing "[CONTROL] C".  Many of the algorithms perform user-friendly range checking, but not data type checking.

 

 

6.1.7        Changing Arithmetic Precision

 

There may be a "slight" difference to the solutions that these algorithms produce as compared to those shown in the text examples.  This is usually a result of different word sizes used in the computations (ie - float, double, long double).  This is a computer and compiler dependant situation and can be expected -- within reason.  Only deviations in the least significant digits should be noticeable.  An accumulation of this round-off error may result in the variation of even more significant digits.  See the header file <float.h> for the expected number of significant digits when using your C compiler.

 

Most digital computers use floating-point formats which provide a close but by no means exact simulation of real number arithmetic.  Among other things, the associative and distributive laws do not hold completely (i.e. order of operation may be important, repeated addition is not necessarily equivalent to multiplication).  Underflow or cumulative precision loss is often a problem.

 

Do not assume that floating-point results will be exact.  These problems are no worse for C than they are for any other computer language.  Floating-point semantics are usually defined as "however the processor does them;" otherwise a compiler for a machine without the "right" model would have to do prohibitively expensive emulations.  More accurate result can usually be obtained by increasing the precision from type "float" to type "double", or from type "double" to type "long double."

 

When changing a program's precision to or from different floating-point types, remember to change the following:

 

               FLOAT          DOUBLE         LONG DOUBLE

Variables:     float          double         long double

 

printf():      %f             %lf            %Lf

               %g or %G       %lg or %lG     %Lg or %LG

               %e or %E       %le or %lE     %Le or %LE

               % f, etc.      % lf, etc.     % Lf, etc.

               %.9f, etc.     %.16lf, etc.   %.25Lf, etc.

 

naautil.c:     vector();      dvector();

               matrix();      dmatrix();

 

naautil2.c:                                   ldvector();

                       ldmatrix();

 

Some C compilers may add an "f" prefix to their math functions to distinguish them as returning float types instead of the usual double type.  These may be implemented as compiler extensions (such as the MIPS C compiler) but are not part of the ANSI C standard.

 

        Float          Double

        fsin();        sin();

        fcos();        cos();

        ftan();        tan();

        fasin();       asin();

        facos();       acos();

        fatan();       atan();

        fsinh();       sinh();

        fcosh();       cosh();

        ftanh();    tanh();

 

 

6.1.8        Using Floating-Point Numbers in Functions

 

When modifying function equations, be sure to type all constants in floating-point format.  Good C compilers know that if one argument in an expression is a floating-point value then all integer types will be promoted (converted) to floating-point values.  There is no guarantee of getting a correct result especially since many older compilers do not implement strong prototyping.

A common error is to type:

 

          return ((3/2)*sin(x));     /* Bad Example */

 

instead of:

 

          return ((3.0/2.0)*sin(x));     /* Good Example */

 

The first expression returns "1*sin(x)" while the later returns "1.5*sin(x)".  The first is incorrect since with C integer arithmetic, 3/2 equals 1, being truncated to the nearest integer.  A "lazy man's" way is to type:

 

          return ((3./2)*sin(x));        /* Good Example */

 

 

6.1.9        The Pow() Function

 

Remember, pow() requires both arguments to be double-precision floating-point types (double).  For  instance, to raise 5.8 to the 3rd power, type "pow(5.8,3.0)" not "pow(5.8,3)".

 

 

6.1.10      Implementing SIG-Digit Rounding/Truncation

 

To modify a program to work with SIG‑digit rounding arithmetic, do the below steps:

 

NOTE:        To implement SIG‑digit truncation or chopping, replace the word "round" with the word "trunc".

 

Example:

#include "round.c" -‑‑> #include "trunc.c"

round(num,SIG) ----‑-‑> trunc(num,SIG)

 

1.       Add the below #include file:

 

                   #include "round.c" /* Rounds X to SIG significant digits. */

 

This file requires <float.h> and <math.h> which are already included inside of "naautil.c."

 

2.       Add to the global variable list, above main() (or locally inside of main() if round() is ONLY used inside main()), the following:

 

                   int SIG;

 

3.       Prompt for the number of significant digits, SIG, using the code:

 

  do {

    printf("Enter the number of significant digits, SIG (1‑%d): ",

      DBL_DIG);

    scanf("%d", &SIG);

    if (SIG < 1 || SIG > DBL_DIG)     /* Range checking */

      printf("Enter 1 to %d only for number of significant digits.\n",

        DBL_DIG);

  } while (SIG < 1 || SIG > DBL_DIG);

  fprintf(file_id, "Computed with %d‑digit rounding arithmetic.\n\n",

    SIG);

 

          NOTE:        DBL_DIG is defined in <float.h> and is usually has the value of around "10".

 

4.       Now, for EVERY number and after EVERY computation (ie‑ +,‑,*,/, pow(), sqrt(), etc.) add a line similar to the following:

 

              num = round(num, SIG);

 

or just "round(num, SIG)" if in the middle of an equation.

 

5.       (OPTIONAL) Change the output line to show only SIG digits using "*" and "SIG", like:

 

    printf("% *g ", SIG, X[i]);

 

6.       (OPTIONAL) Change all doubles to floats and all "%lg", "%lf", and "%le"'s to "%g", "%f", and "%e" as well as all dmatrix() and dvector() to matrix() and vector() as explained in Sub-Section 6.1.7.

 

7.       If the Tolerance is prompted for, like below:

 

        printf("Enter the tolerance, TOL (1.0e‑4): ");

        scanf("%lf", &TOL);

        fprintf(file_id, "Tolerance = %lg\n\n", TOL);

 

replace it with:

 

        TOL = 0.5*pow(10.0, ‑((double) SIG));

        fprintf(file_id, "Tolerance = %lg\n\n", TOL);

 

 

6.1.11      Floating-Point Output Alignment

 

Many of the programs attempt to print out answers in columns, such as for tables (chapters 2, 3, 5, 7‑12) and matrices (chapters 6, 7, 9).  Assuming the majority of the programs would be used for "normally small" numbers, printf() was used with "%g" and "%f" format arguments.  This can causes the output to appear unaligned if large numbers are printed along side small numbers.  If you would like to have the output align all the time then use "%e".  This forces ALL numbers to be of the form:

 

          ‑3.14159e‑002               [sign] [mantissa] e [[sign] exponent]

 

Alignment is guaranteed, but the numbers often take up more room than is necessary and can be less easy to read.

 

 

6.2 Converting Programs into Functions

 

After becoming familiar with several of these algorithms, many users desire to use them as stand-alone functions to be called from within other C programs.  Several books may be purchased which provide only functions, not stand-alone programs, such as the book "Numerical Recipes."  Extra care has been placed into all of the "Numerical Analysis Algorithms in C" programs to help make converting them into functions easier.

 

Modifying these algorithms to be FORTRAN callable is also possible.  The details for this procedure are too detailed and compiler dependent to be listed in this general-purpose User's Manual.

 

Converting a stand-alone algorithm into a function can be simpler than you might think.  Most of the process involves deleting the unnecessary input and output code.  An example using Algorithm 4.1 listed in Appendix A is given for completeness.

 

To convert a stand-alone program into a function, perform the following steps:

 

1.       Rename "main()" to a proper function name, such as "simpson()."  Be sure to place the appropriate return type (usually double) before the function name.  Example:

 

          From:          main()

          To:              double simpson()

 

2.       Separate the variables that follow "main()" into two groups: those to be passed as parameters and those that are internal to the function.  Refer to the INPUT section in the comments at the top of each algorithm to determine the passed parameters.  Place the passed parameters into the function parentheses, such as:

 

        double simpson (a, b, n)       /* K&R Style */

        double a, b;

        int n;

or

    double simpson (double a, double b, int n)  /* ANSI Style */

 

Ensure that the internal variables are placed after the first "{" character.

 

3.       Delete any unnecessary global variables, such as "char *outfile ..." and "char *eq_text_f ..."

 

4.       Replace all function definitions (not calls), such as f(x), with a proper prototype, such as:

 

    double f();    /* K&R Style */

or

    double f(double x); /* ANSI Style */

 

This instructs your C compiler that the function f receives a variable of type double and returns a result of type double.  Failure to do this may cause the function f to return erroneous integer results.

 

5.       Remove most all of the code under the INPUTS section.  You may want to keep any range checking code, such as:

 

        if (n <= 0) {

          printf("ERROR - n must be greater than zero.\n);

          exit (-1);           /* Exit to system */

    }

 

6.       Keep the code under the ALGORITHM section.  This will form the "brains" of the new function.

 

7.       Replace all of the code under the OUTPUTS section with a single return() statement.  The only exception would be to leave any "free_*()" calls.  The return() call should be the last statement of the new function.  The return value should match that in the top comments of the program.  For "041.C", use:

 

    return (XI);

 

8.       Double-check for and remove any unwanted printf() and scanf() routines.  Most mathematical functions do not use them.  Scanf() data should be passed to the function, while printf() output should be handled by the calling main program.

 

 

6.2.1        An Example Using Simpson's Rule

 

Algorithm 4.1 - Composite Simpson's Rule ("041.c") was converted into a stand-alone function named simpson() as shown below.  This function can also be found in the UTIL sub-directory in a file named "041fun.c."

 

/*********************************************************************

               Composite Simpson's Rule ‑ Algorithm 4.1

                       As A Stand‑Alone Function

**********************************************************************

 

                                !b

To approximate the integral I = *  f(x) dx:

                                "a

 

INPUT endpoints a, b; even positive integer n; the function f().

 

OUTPUT approximation XI to I.

 

NOTE: Listed as Simpson's Composite Rule in 3rd edition of the text.

 

**********************************************************************

*  Written by:  Harold A. Toomey, CARE‑FREE SOFTWARE, 3Q 1991, v4.2  *

*********************************************************************/

 

#include "naautil.c"   /* Numerical Analysis Algorithms Utilities. */

double f(double x);            /* Function prototype */

 

 

double simpson (a, b, n)

double a, b;

int n;

{

  double h, X, XI, XI0, XI1, XI2, f();

  int i;

 

  if ((n <= 0) || (n % 2 != 0)) {     /* Range checking */

    printf("ERROR ‑ n must be even and greater than zero.\n");

    exit (‑1);                 /* Exit to system */

  }

 

  /*************

   * ALGORITHM *

   *************/

 

  /* STEP #1 */

  h = (b ‑ a)/n;

 

  /* STEP #2 */

  XI0 = f(a) + f(b);

  XI1 = 0.0;                   /* Summation of f(x(2i‑1)). */

  XI2 = 0.0;                   /* Summation of f(x(2i)).   */

 

  /* STEP #3 */

  for (i=1;i<n;i++) {

 

    /* STEP #4 */

    X = a + i*h;

 

    /* STEP #5 */

    if (i % 2 == 0)

      XI2 += f(X);             /* For even i. */

    else

      XI1 += f(X);             /* For odd i.  */

  }

 

  /* STEP #6 */

  XI = h*(XI0 + 2.0*XI2 + 4.0*XI1) / 3.0;

 

  return (XI);

 

}                              /* STOP */

 

/********************************************************************/

/*  Copyright (C) 1988‑1991, Harold A. Toomey, All Rights Reserved. */

/********************************************************************/

 

 

6.3 Using Input Files (*.IN)

 

An input file is provided in the IN sub-directory for each algorithm.  Each file contains the same name as the algorithm, but with a ".in" extension instead of ".c".  The contents of each input file match the examples given in the text following each algorithm.  They were used to create the accompanying output files for each algorithm in the OUT sub-directory.

 

Input files can be used to save time.  They are especially helpful when working with large arrays where only small changes are made from run to run.  Input files consist of simple text just as you would enter it if the program prompted you for it.

 

Please note that the input files provided with "Numerical Analysis Algorithms in C" require that the below "naautil.c" flags be set as follows:

 

        FLAG            SETTING

        TITLE_PROMPT   TRUE

    EQ_EVAL     FALSE

 

Input files can be redirected as input as a program is run.  For example, to "feed" Algorithm 4.1 with data from an input file, type one of the following:

 

MS-DOS:

          C:\NAA42> 041 < IN\041.IN

 

UNIX:

          % 041 < in/041.in

 

VAX/VMS:

$ DEFINE SYS$INPUT 041.IN  - assumes "041.IN" is in the current directory

$ RUN 041

$ DEASSIGN SYS$INPUT

 

MACINTOSH with THINK C 4.0:

 

To use redirection on a Macintosh with the THINK C 4.0 compiler, each algorithm must be modified as follows:

 

1.       Add these two lines of code just before main():

 

               #include <console.h>

        int ccommand (char ***p);

 

2.       Add arguments (parameters) to main() as shown below:

        main(int argc, char **argv)

 

3.       Just after the variable declarations for main() and before calling "NAA_do_first(outfile);", add:

 

        argc = ccommand(&argv);

 

After making these modifications, Algorithm 4.1 should look like this:

 

...

#include <console.h>

int ccommand (char ***p);

 

main(int argc, char **argv)

{

  double a, b, h, X, XI, XI0, XI1, XI2, f();

  int i, n;

 

  argc = ccommand(&argv);

 

  NAA_do_first(outfile);  /* NAA initialization procedure. */

  ...

    }

 

Be sure to link to the "ANSI" library.  It contains the ccommand() console command.

 

Now, when the modified algorithm is run, a command-line window will appear.  Ensure that the input file is in the same folder as "041.c" and enter: "041 < 041.in".

 

 

6.4 Using Output Files (*.OUT)

 

An output file is provided in the OUT sub-directory for each algorithm.  They contain the same name as the algorithms, but with a ".out" file extension instead of ".c".  The default name of an algorithm's output file can be easily changed by modifying "char *outfile = "nnn.out"; " at the top of each individual algorithm.  The contents of each output file match the examples given in the text following each algorithm.  They were created by redirecting the input files found in the IN sub-directory.

 

These output files can be used to verify that each algorithm is performing as expected.  Use them to compare your output results on your computer system.

 

Output files differ somewhat from what you see when a program is run.  Output files format the output into a more condensed and ready-to-print format.  They are created with calls to printf2() and fprintf(file_id,...) ONLY.  The FILE_SAVE flag in "naautil.c" must be set to TRUE to create output files.

 

 

6.5 Explanation of the Naautil.c File

 

The "naautil.c" file is the most important file of all the "Numerical Analysis Algorithm in C" files.  It contains functions and routines that are used in every algorithm.  It also allows these programs to work on many non-standard C compilers.  The "naautil.c" file should be included in all of the programs using #include "naautil.c".

 

If your C compiler is not truly ANSI C compliant, the "naautil.c" file will be the first to correct it or the first to cause error messages.  The complete source code for "naautil.c" is listed in Appendix B.  This file also contains several flags or #define statements which you can set to get the most out of these algorithms.

 

"Naautil.c" also defines the constant B (PI) . 3.14159..., although it can often be found in some system header files.  It is most useful in trigonometric functions.  The constants "TRUE" and "FALSE" are also defined just in case the system header files fail to define them.

 

 

6.5.1        #Define Flags

 

"Naautil.c" has eight flags that can be set.  Most are usually set only once.  An explanation of each flag is given below.

 

ANSI:

If your compiler supports the ANSI C standard, then set ANSI to TRUE.  Set ANSI to FALSE only if the programs will not compile with it set to TRUE.  This flag is used for strong prototyping of functions.  It is used by all of the supporting ".c" files and in the utilities as well.

 

ANSI_FUNCT:

This flag should be set to TRUE to use ANSI style functions.  Setting it to FALSE should work on truly ANSI compliant compilers as well.  See Section 9.1 for an example.  This flag must be set to TRUE for THINK C 4.0 on a Macintosh.

 

FILE_SAVE:

If you would like to save the output of the algorithms to a file, then set FILE_SAVE to TRUE.  The output is still printed to the screen as you run the program.  Set it to FALSE if you do not plan to save the output to a file.  Used only in the functions printf2(), NAA_do_first(), and NAA_do_last().

 

TITLE_PROMPT:

If you would like to be prompted for an optional title at the start of each program, then set TITLE_PROMPT to TRUE.  This is useful when the output is to be handed in as homework, allowing the user's name or the problem number to be entered.  Hitting the [ENTER] key, instead of text for a title, causes no title to be printed to the output file.  Set it to FALSE if you do not want to be bothered with entering a title every time you run an algorithm.  Used only in the function NAA_do_first().

 

EQ_EVAL:

Several of the algorithms require a single function to be evaluated.  Set EQ_EVAL to TRUE if you wish to enter the function during run-time instead of at compile time.  A couple of simple modifications were made to the algorithms to allow this option to work.  See Chapter 8 - "The Equation Evaluator Routines" for instructions on using this option.

 

When this flag is set to TRUE, the 1000+ line file "eqeval.c" is included into "naautil.c" and compiled with the algorithm.  This flag is used in the function NAA_do_first() as well as in "041ee.c" and "ee.c" in the UTIL sub-directory.

 

NAAUTIL_OBJ:

This option is useful for frequent users who wish to speed up the compilation process.  It should be set to TRUE only if "naautil.c" has been pre-compiled into object code.  See Section 6.6 - "Using Naautil.c as Object Code" for more details.

 

OLD_UNIX_OS:

This flag is only necessary for older UNIX computers which use <varargs.h> instead of <stdarg.h> as the header file for variable length argument lists.  Variable length arguments are used only in printf2() and in "eqeval.c" 's eval_eq() routine.

 

NO_LONG_DOUBLES:

Set this flag to TRUE if you are not using the "long double" type routines for higher precision, or if your compiler does not support the "long double" type.  The "long double" type is used in several routines in "naautil2.c", but is not used in any of the algorithms.  It is provided for the user to obtain more accurate numeric results wherever float of double types are being used.  This flag should be set to TRUE for some VAX C compilers.  Setting this flag to FALSE will compile six routines which take about 1K bytes of object code.

 

 

6.5.2        Flag Default Settings

 

        FLAG           SETTING

        ANSI           TRUE

        ANSI_FUNCT     FALSE       (Is set to TRUE on Macintosh disks)        

        TITLE_PROMPT   TRUE

        FILE_SAVE      TRUE

        EQ_EVAL        FALSE       (Set to TRUE when using "041ee.c")

        NAAUTIL_OBJ    FALSE

        OLD_UNIX_OS    FALSE

        NO_LONG_DOUBLES TRUE

 

        EQEVAL_OBJ  FALSE       (In "eqeval.c" only)

 

 

6.5.3        Description of the Routines

 

The "naautil.c" file contains the following procedures and functions:

 

          Return                  Procedure

          Value          Name                             Description

                   void    naaerror         Exits program with an error message

 

                   double** dmatrix         Allocates a 2‑D array of doubles

                   float** matrix           Allocates a 2‑D array of floats

                   double* dvector          Allocates a 1‑D array of doubles

                   float*  vector           Allocates a 1‑D array of floats

                   int*    ivector          Allocates a 1‑D array of integers

                   void    free_dmatrix     Frees the allocated 2‑D array memory

                   void    free_matrix      Frees the allocated 2‑D array memory

                   void    free_dvector     Frees the allocated 1‑D array memory

                   void    free_vector      Frees the allocated 1‑D array memory

                   void    free_ivector     Frees the allocated 1‑D array memory

 

                   int     printf2          Like printf(), but writes to a file as well

                   void    NAA_do_first     NAA initialization procedure

                   void    NAA_do_last      NAA final procedure

 

Some of these functions can be found in the book "Numerical Recipes in C".  They have been tailored for "Numerical Analysis Algorithms in C."

 

 

naaerror():

 

This Numerical Analysis Algorithms Error handler prints error messages then exits the program to the operating system.  It is used by most of the routines found in "naautil.c", "naautil2.c" and "naautil3.c" as well as in several of the algorithms.

 

 

dmatrix():

 

"Naautil.c" defines five routines for allocating 1-D and 2-D arrays.  These are:

 

          ivector()         Allocates a 1‑D array of integers

          vector()           Allocates a 1‑D array of floats

          dvector()         Allocates a 1‑D array of doubles

          matrix()           Allocates a 2‑D array of floats

          dmatrix()         Allocates a 2‑D array of doubles

 

These routines are often used instead of conventional arrays.  For example:

 

          double **A;

          A = dmatrix(0,9,0,11); /* Dynamic method */

 

replaces

 

          double A[10][12];      /* Array method */

 

These simple routines are used for three reasons: speed, flexibility, and efficiency.

 

Speed:

For the 2-D array above, referencing two pointers (2 adds) to obtain a value is usually faster than using an add and a multiply (1 add + 1 multiply) inherent when indexing arrays.  The array "A" is used identically in both situations.  To obtain this speed, a few more bytes of memory are used to store a row of pointers.

 

Flexibility:

With the array method, the number of elements for each dimension are specified.  The above example uses 10 rows and 12 columns.  These must be referenced from 0 to 9 and 0 to 11 respectively.  With the dmatrix() routine, the RANGES of the elements for each dimension are specified.  This makes it easier to work with arrays which are not referenced from 0 to n-1.  Even negative ranges may be specified, such as dvector(-2,3).

 

For example, assume we need to sum five elements from 5 to 10.  The dvector() routine could be used to allocate storage space as follows:

 

          double *B;

          B = dvector(5,10);

 

The sum of i from 5 to 10 could be easily implemented with:

 

          for (i=5;i<=10;i++)

            sum = sum + B[i];

 

Efficiency:

As seen by the above implementation, B stores only 6 elements.  If we used "double B[6];" (the array method) we would be required to adjust the index, i, or to just declare B with 11 elements "double B[11];" for readability.  This would waste 5 elements!  The matrix and vector routines never waste variables since you only declare what you will use.

 

The matrix and vector routines call calloc() to dynamically allocate memory.  This means a program which operates on an array of n x n elements needs to allocate only n x n elements.  With the array method, the largest anticipated array must be declared which is usually wasteful (consider A[100][100] for a simple 4 x 4 matrix operation!).

 

"Naautil2.c" contain more matrix and vector routines for other variable types.  It also defines cube routines (like dcube()) for 3-D matrices.  These are fast but utilize an extra array of pointers as a trade off.  "Naautil3.c" contain vector, matrix and cube routines for complex data types.

 

If your older C compiler does not have calloc() implemented, use "calloc.c" inside the UTIL sub-directory.  Malloc() could also be used only if every vector, matrix and cube element is initialized to zero before using them in each algorithm.

 

 

free_dmatrix():

 

Every vector, matrix, and cube routine has a free_ routine to match it.  The free_ routines, like free_dmatrix(), de-allocate the memory allocated by the vector, matrix, and cube routines.  These are particularly useful if the algorithms are to be converted into stand-alone functions.  Some older compilers require that the free_ routines be called in reverse order from the vector, matrix, and cube routines which allocated the memory blocks.  This reverse ordering style has been used with all of the algorithms.

 

 

printf2():

 

This simple routine works exactly like printf(), but it sends its output to a file as well.  The output file is the one defined at the top of each algorithm (char *outfile), which gets assigned to the file pointer "file_id."  It is used frequently in the algorithms to make the source code shorter and easier to read.  It uses variable length arguments which are often non-portable to non-ANSI compliant compilers.

 

Two separate versions of this routine are provided.  The first uses <varargs.h> as the header file and is included for older UNIX C compilers.  The second uses <stdarg.h> as the header file and is ANSI compliant.  Only one of these routines can be used at a time.  The OLD_UNIX_OS flag determines which routine is selected, assuming the FILE_SAVE flag is set to TRUE.

 

 

NAA_do_first():

 

This routine is used in every algorithm as the first executable statement.  It performs four main functions and is dependant upon several flag settings:

 

1.                Opens the output file for writing (if FILE_SAVE == TRUE)

2.                Prints the "Numerical Analysis Algorithms in C" banner

3.                Prompts for an optional title (if TITLE_PROMPT == TRUE)

4.       Prompts for the use of the Equation Evaluator routines and gets the equation (if EQ_EVAL == TRUE)

 

 

NAA_do_last():

 

This routine is used in every algorithm as the last executable statement.  It simply closes the output file opened by NAA_do_first() and informs the user that a file has been created.  This routine is used only when the FILE_SAVE flag is set to TRUE.

 

 

6.6 Using Naautil.c as Object Code

 

Each of the algorithms use the file "naautil.c."  Both the "naautil.c" and "eqeval.c" files can be easily compiled into object code once and then used thereafter ("naautil.c" includes "eqeval.c" if the EQ_EVAL flag is set to TRUE).  This can save hours of recompilation time, especially when using many algorithms over a period of time, like for a numerical methods course.  The below sub-sections describe this procedure for different computer systems.  The files described in Section 7.3 - "Time-Saving Batch, Script and Command Files" contain commented-out code to do this as well.

 

Note that if any flags are changed in "naautil.c", then it must be recompiled into object code again before the changes can take effect.  This includes changing the TITLE_PROMPT, FILE_SAVE, and EQ_EVAL flags.

 

 

6.6.1        MS-DOS

 

Object code files in MS-DOS have a ".OBJ" extension.  To create object code, do the following:

 

1.       Set the NAAUTIL_OBJ flag to FALSE in "naautil.c"

 

2.       Compile "naautil.c" into object code by typing the following command at the DOS prompt:  (assumes Microsoft C 5.0)

 

    C:\NAA42> CL /c NAAUTIL.C

 

3.       Set the NAAUTIL_OBJ flag back to TRUE in "naautil.c"

4.       From now on, compile the algorithms into object code, then link "naautil.obj" to them.  For example, using "041.c", type:

 

        C:\NAA42> CL /c 041.C

    C:\NAA42> CL 041 NAAUTIL

 

The first command creates "041.obj" while the second command links it to the "naautil.obj" object file to form the executable "041.exe."

 

 

6.6.2        UNIX

 

Object code files for UNIX have a ".o" extension.  To create object code, do the following:

 

1.       Set the NAAUTIL_OBJ flag to FALSE in "naautil.c"

 

2.       Compile "naautil.c" into object code by typing the following at the shell prompt:

 

    % cc ‑c naautil.c

 

3.       Set the NAAUTIL_OBJ flag back to TRUE in "naautil.c"

 

4.       From now on, compile the algorithms along with "naautil.o." For example, using "041.c", type:

 

    % cc 041.c ‑o 041 naautil.o ‑lm

 

 

6.6.3        Macintosh

 

Object code files for THINK C on a Macintosh are indicated in the project window by a non-zero size after the source file's name.  To create the object code, do the following:

 

1.       Set the NAAUTIL_OBJ flag to FALSE in "naautil.c"

 

2.       Compile "naautil.c" into object code.

 

3.       Set the NAAUTIL_OBJ flag back to TRUE in "naautil.c"

 

4.       From now on, compile the algorithm into object code, then link the "naautil.c" object code to it.

 

You may have trouble if the compiler asks to bring the "naautil.c" file up to date after step #3 above.  This may happen since setting the NAAUTIL_OBJ flag back to TRUE in "naautil.c" marks it as no longer current.  Bringing the folder up to date, including "naautil.c", would remove the routines from the object code compiled in step #2.

 

 

6.6.4        VAX/VMS

 

Object code files for VAX/VMS have a ".OBJ" extension.  To create the object code, do the following:

1.       Set the NAAUTIL_OBJ flag to FALSE in "naautil.c"

 

2.       Compile "naautil.c" into object code by typing the following at the VMS prompt:

 

    $ CC /G_FLOAT NAAUTIL.C

 

3.       Set the NAAUTIL_OBJ flag back to TRUE in "naautil.c"

 

4.       From now on, compile the algorithm into object code, then link "naautil.obj" to it.  For example, using "041.c", type:

 

        $ CC /G_FLOAT 041.C

    $ LINK 041, NAAUTIL, LNK$LIBRARY/LIB, LNK$LIBRARY_1/LIB

 

The first command creates "041.obj" while the second command links it to the "naautil.obj" object file to form the executable "041.exe."

 

 

6.7 Supporting C Source Code Usage List

 

The list below outlines the support files used by each chapter:

 

                                COMPLEX.C             ROUND.C

                                   and                  and

Chapter  NAAUTIL.C  NAAUTIL2.C  NAAUTIL3.C  GAUSSJ.C  TRUNC.C EQEVAL.C

  1         X                                                    X

  2         X                       X                            X

  3         X                                          X       X

  4         X                                            X       X

  5         X                                                    X

  6         X                                  X         X

  7         X                                           

  8         X           X           X          X                 X

  9         X                                  X        

  10        X                                  X

  11        X                                  X

  12        X           X                                        X

 

File usage by name:

 

          NAAUTIL.C           - All .C files

          NAAUTIL2.C         - 081.C and 125.C   

          NAAUTIL3.C         - 027.C, 028A.C, and 081.C

          COMPLEX.C        - Used in NAAUTIL3.C only

          GAUSSJ.C           - 060B.C, 080B.C, 093.C, 101.C, 101A.C, 102.C, 116.C, 125.C and 129A.C

          ROUND.C            - 031B.C, 040D2.C, 061B.C, 061C2.C, 061D2.C, 062B.C, 063B.C, 074.C, and 095D.C

          TRUNC.C             - Not used.  May replace ROUND.C in the homework exercises for chopping arithmetic.

          EQEVAL.C           - See Section 8.8 for a list.

 

 

6.8 "Numerical Analysis" Text Errors and Corrections

 

This section lists a few errors encountered in the texts as the algorithms were being programmed into C.  Many of the algorithms will not work correctly without these corrections.  The errors are listed separately for the third and fourth editions of the text.  Perhaps a more complete list of corrections may be obtained from the publisher, PWS-Kent Publishing Company, 20 Park Plaza, Boston, Massachusetts  02116.

 

 

6.8.1        3rd Edition Errors

 

TEXT ERRORS AND CORRECTIONS

for

"Numerical Analysis", third edition,

Richard L. Burden & J. Douglas Faires, 1984

 

 

Page#   Location       Fix

‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑

345     Algorithm 6.5  In Step 7, only swap a few elements in matrix L.  Use: For k = 1 to i‑1, swap L(p,k) with L(i,k).  This does not apply to matrix A.