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
CareFree Software
3rd Quarter 1991
Technical Publications:
Harold
A. Toomey
Programming:
Harold
A. Toomey
© Copyright 19881993, 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 930742CUM2
CareFree Software
Attn: Harold Allen Toomey
1376 N. 1100 E.
American Fork, UT
84003
18014921526
IBM is a trademark of International Business Machines
Corporation.
Microsoft and MSDOS 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. StepByStep Examples on Various Computers
4‑1
4.1 Need
List
4‑1
4.2 Customizing
Naautil.c 4‑1
4.3 Example
Using MSDOS, Microsoft C and the PEdit 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
FloatingPoint Numbers in Functions 6‑3
6.1.9 The
Pow() Function 6‑4
6.1.10 Implementing
SIGDigit Rounding/Truncation 6‑4
6.1.11 FloatingPoint
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 MSDOS 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 RunTime 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 FloatingPoint
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 TimeSaving
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 PreCompiled 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 TradeOffs 8‑6
9. Portability 9‑1
9.1 C
vs ANSI C 9‑2
9.2 IBM
PCs and MSDOS 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 MSDOS 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 standalone 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  Initialvalue 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  Boundaryvalue 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,
leastsquare 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 
"StepByStep 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 SubSection 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 generalpurpose 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 generalpurpose
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 generalpurpose 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 CAREFREE SOFTWARE at 18017850464. 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 CareFree Software have had this step done already.
4. It is
recommended that the algorithms be placed in their own subdirectory (or
Macintosh folder), such as "naa42."
This subdirectory can be created and entered by typing one of the
following sets of commands:
MSDOS:
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 selfdocumenting. 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 CAREFREE 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 wellsupported 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")
Kermit32>
set file_type binary (or:
set
file_type ascii)
Kermit32>
receive
For a UNIX workstation, type:
% kermit
Kermit32>
set binary (or: set ascii)
Kermit32>
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.
Kermit32>
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
standalone 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 MSDOS 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 subdirectory in
files named with a ".out" extension.
Their inputs are found in the IN subdirectory in files named with a
".in" extension.
Over twothirds 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
runtime. 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"
GaussJordan 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 floatingpoint value to SIG
significant digits
TRUNC.C  Truncates a floatingpoint 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 BinarySearch) Algorithm
2.1
022.C*  FixedPoint Algorithm
2.2
023.C  NewtonRaphson Algorithm
2.3
024.C*  Secant Algorithm
2.4
024B.C* 
Method of False Position (or Regula Falsi) Algorithm 2.4B
024C.C 
Modified NewtonRaphson 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 DividedDifference 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* 
NewtonCotes 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 InitialValue 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*  RungeKutta (Order Four) Algorithm
5.2
053.C  RungeKuttaFehlberg Algorithm
5.3
054.C*  Adam's FourthOrder PredictorCorrector Algorithm 5.4
054B.C* 
AdamsBashforth (all four) and Milne's Methods Algorithm 5.4B
054C.C* 
MilneSimpson PredictorCorrector Algorithm
5.4C
055.C*  Adam's Variable StepSize PredictorCorrector Algorithm 5.5
056.C* + Extrapolation Algorithm
5.6
057.C  RungeKutta for Systems of Differential Equations Algorithm 5.7
057B.C 
Euler's Variable StepSize 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*  GaussJordan Method Algorithm
6.1C1
061C2.C*  GaussJordan Method (with rounding) Algorithm 6.1C2
061D1.C*  GaussianElimination  GaussJordan
Hybrid Method Algorithm 6.1D1
061D2.C*  GaussianElimination  GaussJordan
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*  LDL^{t} Factorization Algorithm
6.5
065B.C* 
LDL^{t} 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*  GaussSeidel 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 (singleprecision) Algorithm
7.4B
CHAPTER 8 Approximation Theory
080B.C* 
LeastSquares 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 Delta^{2} 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 NonSymmetric 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 BoundaryValue 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 RayleighRitz Algorithm
11.5
116.C  Cubic Spline RayleighRitz Algorithm
11.6
CHAPTER 12 Numerical Solutions to PartialDifferential
Equations
121.C  Poisson Equation FiniteDifference (Elliptic) Algorithm
12.1
122.C*  Heat Equation BackwardDifference (Parabolic) Algorithm 12.2
122B.C* 
Heat Equation ForwardDifference (Parabolic) Algorithm 12.2B
122C.C* 
Heat Equation Richardson's Method (Parabolic) Algorithm 12.2C
123.C*  CrankNicolson (Parabolic) Algorithm
12.3
124.C  Wave Equation FiniteDifference (Hyperbolic) Algorithm 12.4
125.C  FiniteElement Algorithm
12.5
126A.C  Parabolic Equations With Newton Iteration in 1D Algorithm 12.6A
127A.C  Parabolic Equations With Newton Iteration in 2D Algorithm 12.7A
128A.C  Elliptic Equations With Newton Iteration in 2D Algorithm 12.8A
129A.C  Biharmonic Equation Using GaussJordan 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 runtime. 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 GaussJordan 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 lessused 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 floatingpoint value to
SIG significant digits. Only 9
algorithms currently use this function.
See SubSection 6.1.10 to see how this file is used.
TRUNC.C
"Trunc.c" truncates, or chops, a
floatingpoint 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 MSDOS'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 standalone function.
CONVERT.C
"Convert.c" is the C source code for a
utility which translates text files into standard sevenbit ASCII files. It is useful before placing these algorithms
on nonMSDOS 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 MSDOS 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 subdirectory 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 MSDOS 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 MSDOS 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 viiviii 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. StepByStep Examples on Various Computers
This chapter gives four stepbystep 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 twothirds of the algorithms, Steps 24 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: MSDOS
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 runtime 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 MSDOS, Microsoft C and the
PEdit Editor
This example uses the following software:
Operating
System: MSDOS on an IBM PC
Compiler: Microsoft C 5.0
Editor:
WordPerfect's
PEdit Editor
No special "naautil.c" flags need to be set.
This example assumes the files were installed onto the
"C" drive in the "\NAA42" subdirectory. 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" subdirectory 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
textonly 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 SubSection 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
readytoprint 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" subdirectory 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" subdirectory, 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
floatingpoint results.
The script file "ccc" can also be used in
place of the "cc" command.
See SubSection 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
readytoprint 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
"ANSIsmall" or "ANSIA4" since they do not support
floatingpoint operations. If you have
a math coprocessor (MC68881), substitute "ANSI" with
"ANSI881". 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
readytoprint 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 80bit 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"
(AddisonWesley).
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" subdirectory, 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 fullscreen 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 floatingpoint 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 online 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 SubSection 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 = 5.000000e02 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
readytoprint 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 18003330330, or write to: MIX Software, 1132 Commerce Dr.,
Richardson, TX 75081, (214) 7836001.
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 (pre1987), 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 prestandard 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
builtin 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 floatingpoint 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
subdirectory 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 onethird 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 subsections 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 1D
127A.c  Parabolic Equations With Newton
Iteration in 2D
128A.c  Elliptic Equations With Newton
Iteration in 2D
129A.c  Biharmonic Equation Using GaussJordan
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
SIGdigit 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 writeprotected 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 floatingpoint 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 userfriendly 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 roundoff 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 floatingpoint 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 floatingpoint results will be
exact. These problems are no worse for
C than they are for any other computer language. Floatingpoint 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 floatingpoint 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 FloatingPoint Numbers in
Functions
When modifying function equations, be sure to type all
constants in floatingpoint format.
Good C compilers know that if one argument in an expression is a
floatingpoint value then all integer types will be promoted (converted) to
floatingpoint 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
doubleprecision floatingpoint 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 SIGDigit 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
SubSection 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 FloatingPoint 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 standalone functions to be called
from within other C programs. Several
books may be purchased which provide only functions, not standalone 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
generalpurpose User's Manual.
Converting a standalone 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 standalone 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. Doublecheck
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 standalone function named simpson()
as shown below. This function can also
be found in the UTIL subdirectory 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 subdirectory 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 subdirectory.
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:
MSDOS:
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
commandline 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 subdirectory
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 subdirectory.
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 readytoprint 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 nonstandard 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 runtime 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 subdirectory.
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 precompiled 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 1D and 2D 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 2D 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 n1. 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 3D 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 subdirectory. 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(), deallocate the memory allocated by the vector,
matrix, and cube routines. These are
particularly useful if the algorithms are to be converted into standalone
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 nonportable to
nonANSI 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
subsections describe this procedure for different computer systems. The files described in Section 7.3 
"TimeSaving Batch, Script and Command Files" contain commentedout
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 MSDOS
Object code files in MSDOS 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 nonzero 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, PWSKent 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.