Numerical
Analysis
Algorithms
in C
Version
4.2
|
Examples
Book
For
"Numerical
Analysis", fourth edition
Richard
L. Burden and J. Douglas Faires
1988
Written
by:
Harold
A. Toomey, MSEE
Care-Free Software
3rd
Quarter 1991
Technical
Publications:
Harold A. Toomey
Programming:
Harold A. Toomey
©
Copyright 1988-1993, Harold A. Toomey - All rights reserved
This document contains proprietary information of
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part, without prior written consent of Harold A. Toomey. Limited rights exist for individual and
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may be used or copied only in accordance with the terms of the license
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this software. See the sample license
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The information in this document is subject to change
without notice.
"Numerical
Analysis Algorithms in C" Examples Book
Version
4.2
Document
Number 9307-42C-EB2
Care-Free Software
Attn:
Harold Allen Toomey
464
North 750 East
Lindon,
UT 84042
1-801-785-0464
Table
of Contents
CHAPTER
1 Mathematical Preliminaries
COMPLEX.C - "Numerical Recipes in C" Complex
Number Routines
EQEVAL.C - Equation Evaluator Routines
GAUSSJ.C - "Numerical Recipes in C"
Gauss-Jordan Matrix Solver
NAAUTIL.C - "Numerical Analysis Algorithms in
C" Utilities I (standard)
NAAUTIL2.C - "Numerical Analysis Algorithms in
C" Utilities II (extended)
NAAUTIL3.C - "Numerical Analysis Algorithms in
C" Utilities III (complex)
ROUND.C - Rounds a floating point value to SIG
significant digits
TRUNC.C - Truncates a floating point value to
SIG significant digits
011B.C* - Taylor Polynomial Approximation Algorithm
1.1B
CHAPTER
2 Solutions of Equations in One
Variable
021.C* -
Bisection (or Binary-Search) Algorithm
2.1
022.C* -
Fixed-Point Algorithm
2.2
023.C - Newton-Raphson Algorithm
2.3
024.C* -
Secant Algorithm
2.4
024B.C* -
Method of False Position (or Regula Falsi) Algorithm 2.4B
024C.C -
Modified Newton-Raphson Method Algorithm
2.4C
025.C* -
Steffensen Algorithm
2.5
026.C* -
Horner Algorithm
2.6
027.C* -
Müller Algorithm
2.7
028A.C* +
Complex Polynomial Solver (CPOLY) Algorithm
2.8A
CHAPTER
3 Interpolation and Polynomial
Approximation
031.C* -
Neville's Iterated Interpolation Algorithm
3.1
031B.C* -
Neville's Iterated Interpolation (with rounding) Algorithm 3.1B
031C.C* -
Aitken's Iterated Interpolation Algorithm
3.1C
032.C* -
Newton's Interpolatory Divided-Difference Formula Algorithm 3.2
033.C* -
Hermite Interpolation Algorithm
3.3
034.C* -
Natural Cubic Spline Algorithm
3.4
035.C* -
Clamped Cubic Spline Algorithm
3.5
CHAPTER
4 Numerical Differentiation and
Integration
040B1.C -
1st Derivative Approximation (for functions) Algorithm 4.0B1
040B2.C* -
1st Derivative Approximation (for tabulated data) Algorithm 4.0B2
040B3.C -
1st Derivative Approximation (for functions w/TOL) Algorithm 4.0B3
040C1.C -
2nd Derivative Approximation (for functions) Algorithm 4.0C1
040C2.C* -
2nd Derivative Approximation (for tabulated data) Algorithm 4.0C2
040D1.C* -
Richardson's Extrapolation Algorithm
4.0D1
040D2.C* -
Richardson's Extrapolation (with rounding) Algorithm 4.0D2
041.C* -
Composite Simpson's Rule Algorithm
4.1
041B.C* -
Composite Trapezoidal Rule Algorithm
4.1B
041C.C* -
Composite Midpoint Rule Algorithm
4.1C
041D.C* -
Newton-Cotes Formulas for Integrals (8 total) Algorithm 4.1D
042.C* -
Adaptive Quadrature Algorithm
4.2
043.C* -
Romberg Algorithm
4.3
043B.C* -
Gaussian Quadrature Algorithm
4.3B
044.C - Composite Simpson's Rule for Double Integrals Algorithm 4.4
044B.C -
Composite Trapezoid Rule for Double Integrals Algorithm 4.4B
044C.C -
Gaussian Quadrature for Double Integrals Algorithm
4.4C
045.C - Composite Simpson's Rule for Triple Integrals Algorithm 4.5
045B.C -
Composite Trapezoid Rule for Triple Integrals Algorithm 4.5B
045C.C -
Gaussian Quadrature for Triple Integrals Algorithm
4.5C
CHAPTER
5 Initial-Value Problems for Ordinary
Differential Equations
051.C* -
Euler Algorithm
5.1
051B.C* -
Midpoint, Modified Euler, and Heun's Methods Algorithm 5.1B
052.C* -
Runge-Kutta (Order Four) Algorithm
5.2
053.C - Runge-Kutta-Fehlberg Algorithm
5.3
054.C* -
Adam's Fourth-Order Predictor-Corrector Algorithm
5.4
054B.C* -
Adams-Bashforth (all four) and Milne's Methods Algorithm 5.4B
054C.C* -
Milne-Simpson Predictor-Corrector Algorithm
5.4C
055.C* -
Adam's Variable Step-size Predictor-Corrector Algorithm 5.5
056.C* +
Extrapolation Algorithm
5.6
057.C - Runge-Kutta for Systems of Differential
Equations Algorithm 5.7
057B.C -
Euler's Variable Step-Size for Systems Algorithm
5.7B
058.C - Trapezoidal with Newton Iteration Algorithm
5.8
CHAPTER
6 Direct Methods for Solving Linear
Systems
060B.C* -
Matrix Inverter Algorithm
6.0B
060C.C* -
Determinant of a Matrix Algorithm
6.0C
060D.C* -
Matrix Multiplier Algorithm
6.0D
061.C* -
Gaussian Elimination with Backward Substitution Algorithm 6.1
061B.C* -
Gaussian Elimination with Backward Substitution Algorithm 6.1B
(with rounding)
061C1.C* - Gauss-Jordan Method Algorithm
6.1C1
061C2.C* - Gauss-Jordan Method (with rounding) Algorithm 6.1C2
061D1.C* - Gaussian-Elimination - Gauss-Jordan
Hybrid Method Algorithm 6.1D1
061D2.C* - Gaussian-Elimination - Gauss-Jordan
Hybrid Method Algorithm 6.1D2
(with rounding)
062.C* -
Gaussian Elimination with Maximal Column Pivoting Algorithm 6.2
062B.C* -
Gaussian Elimination with Maximal Column Pivoting Algorithm 6.2B
(with rounding)
063.C* -
Gaussian Elimination with Scaled Column Pivoting Algorithm 6.3
063B.C* -
Gaussian Elimination with Scaled Column Pivoting Algorithm 6.3B
(with rounding)
064.C* -
Direct Factorization Algorithm
6.4
064B.C* -
Direct Factorization which solves AX=B Algorithm
6.4B
064C.C* -
Direct Factorization with Maximal Column Pivoting Algorithm 6.4C
(3rd edition)
065.C* -
LDLt Factorization Algorithm
6.5
065B.C* -
LDLt Factorization which solves AX=B Algorithm
6.5B
066.C* -
Choleski Algorithm
6.6
066B.C* -
Choleski which solves AX=B Algorithm
6.6B
067.C* -
Crout Reduction for Tridiagonal Linear Systems Algorithm 6.7
CHAPTER
7 Iterative Techniques in Matrix
Algebra
070B.C* -
Vector and Matrix Norms Algorithm
7.0B
071.C* -
Jacobi Iterative Algorithm
7.1
072.C* -
Gauss-Seidel Iterative Algorithm
7.2
073.C* -
Successive Over Relaxation (SOR) Algorithm
7.3
074.C* -
Iterative Refinement (with rounding) Algorithm
7.4
074B.C* -
Iterative Refinement (single-precision) Algorithm
7.4B
CHAPTER
8 Approximation Theory
080B.C* -
Least-Squares Polynomial Approximation Algorithm
8.0B
081.C* +
Fast Fourier Transformation Algorithm
8.1
CHAPTER
9 Approximating Eigenvalues
091.C* -
Power Method Algorithm
9.1
091B.C* -
Power Method with Aitken's 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 Algorithm 9.4B
lambda1
095.C* -
Householder Method Algorithm
9.5
095B.C* -
Householder Method (3rd edition) Algorithm
9.5B
095C.C* -
Householder Method for Non-Symmetric Matrices Algorithm 9.5C
(Upper Hessenberg)
095D.C* -
Householder Method (with rounding) Algorithm
9.5D
096.C* -
QR Algorithm Algorithm
9.6
096B.C* -
QL Algorithm (3rd edition) Algorithm
9.6B
CHAPTER
10 Numerical Solutions of Nonlinear
Systems of Equations
101.C - Newton's Method for Systems Algorithm
10.1
101A.C -
Steffensen's Method for Systems Algorithm
10.1A
102.C - Broyden's Method for Systems Algorithm
10.2
103.C - Steepest Descent Method (with F(x) and J(x)) Algorithm 10.3
103B.C -
Steepest Descent Method (with G(x) and gradG(x)) Algorithm 10.3B
CHAPTER
11 Boundary-Value Problems for Ordinary
Differential Equations
111.C - Linear Shooting Algorithm
11.1
112.C - Nonlinear Shooting with Newton's Method Algorithm 11.2
112B.C -
Nonlinear Shooting with Secant Method Algorithm
11.2B
113.C - Linear Finite Difference Algorithm
11.3
113B.C -
Linear Finite Difference (Richardson's Extrapolation) Algorithm 11.3B
114.C - Nonlinear Finite Difference Algorithm
11.4
114B.C -
Nonlinear Finite Difference (Richardson's Extrap.) Algorithm 11.4B
115.C - Piecewise Linear Rayleigh-Ritz Algorithm
11.5
116.C - Cubic Spline Rayleigh-Ritz Algorithm
11.6
CHAPTER
12 Numerical Solutions to
Partial-Differential Equations
121.C - Poisson Equation Finite-Difference (Elliptic) Algorithm 12.1
122.C* -
Heat Equation Backward-Difference (Parabolic) Algorithm 12.2
122B.C* -
Heat Equation Forward-Difference (Parabolic) Algorithm 12.2B
122C.C* -
Heat Equation Richardson's Method (Parabolic) Algorithm 12.2C
123.C* -
Crank-Nicolson (Parabolic) Algorithm
12.3
124.C - Wave Equation Finite-Difference (Hyperbolic) Algorithm 12.4
125.C - Finite-Element Algorithm
12.5
126A.C -
Parabolic Equations With Newton Iteration in 1-D Algorithm 12.6A
127A.C -
Parabolic Equations With Newton Iteration in 2-D Algorithm 12.7A
128A.C -
Elliptic Equations With Newton Iteration in 2-D Algorithm 12.8A
129A.C -
Biharmonic Equation Using Gauss-Jordan Method Algorithm 12.9A
APPENDIX
A "Numerical Analysis Algorithms in
C" Supporting C Source Code
041EE.C* - Composite Simpson's Rule Using the
Equation Evaluator Routines
041FUN.C - Composite Simpson's Rule as a Function
CONVERT.C* - Converts Files from 8-bit Extended ASCII to
7-bit Standard ASCII
EE.C* - Command-Line Calculator using
the Equation Evaluator Routines
This Examples Book contains the C source code for over
a hundred algorithms found in the text Numerical Analysis, fourth
edition, Richard L. Burden and J. Douglas Faires. The algorithms include listings of the C
source code as well as the accompanying output files for the examples problems
given in the text. The inputs can be
easily extracted from the output listings or can be obtained from the diskettes. Other needed programs and routines are
printed in the appendix.
The '+'s above mean the program may need a larger
stack when compiled and linked.
The '*'s after the above algorithms mean the programs
need to be compiled only once. After the
initial compilation, the executable can be used over and over again.
See the User's Manual for a detailed description on
how to use these algorithms most effectively.
The header file "naautil.c" is included
inside every algorithm. This is the only
file which may need to be modified. It
contains some flags which can be set to make the code more portable for various
compilers. It also contains flags which
allow the user to turn on or off the following options:
1. Prompt for an optional title
2. Save output to a file
3. Use the Equation Evaluator Routines
The following compilers have successfully run all of
the "Numerical Analysis Algorithms in C" v4.2 programs:
1. Microsoft C 5.0 for MS-DOS on an IBM PC
2. THINK C 4.0 on a Macintosh SE (Set ANSI_FUNCT == TRUE)
3. MIPS C compiler (IRIX 3.3.1) on a Silicon
Graphics 4D workstation
4. VAX C v3.1 and v3.2 compilers on a VAX
CHAPTER
1
Mathematical
Preliminaries
|
CHAPTER
1 Mathematical Preliminaries
COMPLEX.C - "Numerical Recipes in C" Complex
Number Routines
EQEVAL.C - Equation Evaluator Routines
GAUSSJ.C - "Numerical Recipes in C"
Gauss-Jordan Matrix Solver
NAAUTIL.C - "Numerical Analysis Algorithms in
C" Utilities I (standard)
NAAUTIL2.C - "Numerical Analysis Algorithms in
C" Utilities II (extended)
NAAUTIL3.C - "Numerical Analysis Algorithms in
C" Utilities III (complex)
ROUND.C - Rounds a floating point value to SIG
significant digits
TRUNC.C - Truncates a floating point value to
SIG significant digits
011B.C* - Taylor Polynomial Approximation Algorithm
1.1B
CHAPTER
2
Solutions
of
Equations
in
One Variable
|
CHAPTER
2 Solutions of Equations in One Variable
021.C* -
Bisection (or Binary-Search) Algorithm
2.1
022.C* -
Fixed-Point Algorithm
2.2
023.C - Newton-Raphson Algorithm
2.3
024.C* -
Secant Algorithm
2.4
024B.C* -
Method of False Position (or Regula Falsi) Algorithm 2.4B
024C.C -
Modified Newton-Raphson Method Algorithm
2.4C
025.C* -
Steffensen Algorithm
2.5
026.C* -
Horner Algorithm
2.6
027.C* -
Müller Algorithm
2.7
028A.C* +
Complex Polynomial Solver (CPOLY) Algorithm
2.8A
CHAPTER
3
Interpolation
and
Polynomial
Approximation
|
CHAPTER
3 Interpolation and Polynomial
Approximation
031.C* -
Neville's Iterated Interpolation Algorithm
3.1
031B.C* -
Neville's Iterated Interpolation (with rounding) Algorithm 3.1B
031C.C* -
Aitken's Iterated Interpolation Algorithm
3.1C
032.C* -
Newton's Interpolatory Divided-Difference Formula Algorithm 3.2
033.C* -
Hermite Interpolation Algorithm
3.3
034.C* -
Natural Cubic Spline Algorithm
3.4
035.C* -
Clamped Cubic Spline Algorithm
3.5
CHAPTER
4
Numerical
Differentiation
and
Integration
|
CHAPTER
4 Numerical Differentiation and
Integration
040B1.C -
1st Derivative Approximation (for functions) Algorithm 4.0B1
040B2.C* -
1st Derivative Approximation (for tabulated data) Algorithm 4.0B2
040B3.C -
1st Derivative Approximation (for functions w/TOL) Algorithm 4.0B3
040C1.C -
2nd Derivative Approximation (for functions) Algorithm 4.0C1
040C2.C* -
2nd Derivative Approximation (for tabulated data) Algorithm 4.0C2
040D1.C* -
Richardson's Extrapolation Algorithm
4.0D1
040D2.C* -
Richardson's Extrapolation (with rounding) Algorithm 4.0D2
041.C* -
Composite Simpson's Rule Algorithm
4.1
041B.C* -
Composite Trapezoidal Rule Algorithm
4.1B
041C.C* -
Composite Midpoint Rule Algorithm
4.1C
041D.C* -
Newton-Cotes Formulas for Integrals (8 total) Algorithm 4.1D
042.C* -
Adaptive Quadrature Algorithm
4.2
043.C* -
Romberg Algorithm
4.3
043B.C* -
Gaussian Quadrature Algorithm
4.3B
044.C - Composite Simpson's Rule for Double Integrals Algorithm 4.4
044B.C -
Composite Trapezoid Rule for Double Integrals Algorithm 4.4B
044C.C -
Gaussian Quadrature for Double Integrals Algorithm
4.4C
045.C - Composite Simpson's Rule for Triple Integrals Algorithm 4.5
045B.C -
Composite Trapezoid Rule for Triple Integrals Algorithm 4.5B
045C.C -
Gaussian Quadrature for Triple Integrals Algorithm
4.5C
CHAPTER
5
Initial-Value
Problems
for
Ordinary
Differential
Equations
|
CHAPTER
5 Initial-Value Problems for Ordinary
Differential Equations
051.C* -
Euler Algorithm
5.1
051B.C* -
Midpoint, Modified Euler, and Heun's Methods Algorithm 5.1B
052.C* -
Runge-Kutta (Order Four) Algorithm
5.2
053.C - Runge-Kutta-Fehlberg Algorithm
5.3
054.C* -
Adam's Fourth-Order Predictor-Corrector Algorithm
5.4
054B.C* -
Adams-Bashforth (all four) and Milne's Methods Algorithm 5.4B
054C.C* -
Milne-Simpson Predictor-Corrector Algorithm
5.4C
055.C* -
Adam's Variable Step-size Predictor-Corrector Algorithm 5.5
056.C* +
Extrapolation Algorithm
5.6
057.C - Runge-Kutta for Systems of Differential
Equations Algorithm 5.7
057B.C -
Euler's Variable Step-Size for Systems Algorithm
5.7B
058.C - Trapezoidal with Newton Iteration Algorithm
5.8
CHAPTER
6
Direct
Methods
for
Solving
Linear
Systems
|
CHAPTER
6 Direct Methods for Solving Linear
Systems
060B.C* -
Matrix Inverter Algorithm
6.0B
060C.C* -
Determinant of a Matrix Algorithm
6.0C
060D.C* -
Matrix Multiplier Algorithm
6.0D
061.C* -
Gaussian Elimination with Backward Substitution Algorithm 6.1
061B.C* -
Gaussian Elimination with Backward Substitution Algorithm 6.1B
(with rounding)
061C1.C* - Gauss-Jordan Method Algorithm
6.1C1
061C2.C* - Gauss-Jordan Method (with rounding) Algorithm 6.1C2
061D1.C* - Gaussian-Elimination - Gauss-Jordan
Hybrid Method Algorithm 6.1D1
061D2.C* - Gaussian-Elimination - Gauss-Jordan
Hybrid Method Algorithm 6.1D2
(with rounding)
062.C* -
Gaussian Elimination with Maximal Column Pivoting Algorithm 6.2
062B.C* -
Gaussian Elimination with Maximal Column Pivoting Algorithm 6.2B
(with rounding)
063.C* -
Gaussian Elimination with Scaled Column Pivoting Algorithm 6.3
063B.C* -
Gaussian Elimination with Scaled Column Pivoting Algorithm 6.3B
(with rounding)
064.C* -
Direct Factorization Algorithm
6.4
064B.C* -
Direct Factorization which solves AX=B Algorithm
6.4B
064C.C* - Direct Factorization with Maximal Column
Pivoting Algorithm 6.4C
(3rd edition)
065.C* -
LDLt Factorization Algorithm
6.5
065B.C* -
LDLt Factorization which solves AX=B Algorithm
6.5B
066.C* -
Choleski Algorithm
6.6
066B.C* -
Choleski which solves AX=B Algorithm
6.6B
067.C* -
Crout Reduction for Tridiagonal Linear Systems Algorithm 6.7
CHAPTER
7
Iterative
Techniques
in
Matrix Algebra
|
CHAPTER
7 Iterative Techniques in Matrix
Algebra
070B.C* -
Vector and Matrix Norms Algorithm
7.0B
071.C* -
Jacobi Iterative Algorithm
7.1
072.C* -
Gauss-Seidel Iterative Algorithm
7.2
073.C* -
Successive Over Relaxation (SOR) Algorithm
7.3
074.C* -
Iterative Refinement (with rounding) Algorithm
7.4
074B.C* -
Iterative Refinement (single-precision) Algorithm
7.4B
CHAPTER
8
Approximation
Theory
|
CHAPTER
8 Approximation Theory
080B.C* -
Least-Squares Polynomial Approximation Algorithm
8.0B
081.C* +
Fast Fourier Transformation Algorithm
8.1
CHAPTER
9
Approximating
Eigenvalues
|
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 Algorithm 9.4B
lambda1
095.C* -
Householder Method Algorithm
9.5
095B.C* -
Householder Method (3rd edition) Algorithm
9.5B
095C.C* -
Householder Method for Non-Symmetric Matrices Algorithm 9.5C
(Upper Hessenberg)
095D.C* -
Householder Method (with rounding) Algorithm
9.5D
096.C* -
QR Algorithm Algorithm
9.6
096B.C* -
QL Algorithm (3rd edition) Algorithm
9.6B
CHAPTER
10
Numerical
Solutions
of
Nonlinear Systems
of
Equations
|
CHAPTER
10 Numerical Solutions of Nonlinear
Systems of Equations
101.C - Newton's Method for Systems Algorithm
10.1
101A.C -
Steffensen's Method for Systems Algorithm
10.1A
102.C - Broyden's Method for Systems Algorithm
10.2
103.C - Steepest Descent Method (with F(x) and J(x)) Algorithm 10.3
103B.C -
Steepest Descent Method (with G(x) and gradG(x)) Algorithm 10.3B
CHAPTER
11
Boundary-Value
Problems
for Ordinary
Differential
Equations
|
CHAPTER
11 Boundary-Value Problems for Ordinary Differential
Equations
111.C - Linear Shooting Algorithm
11.1
112.C - Nonlinear Shooting with Newton's Method Algorithm 11.2
112B.C -
Nonlinear Shooting with Secant Method Algorithm
11.2B
113.C - Linear Finite Difference Algorithm
11.3
113B.C -
Linear Finite Difference (Richardson's Extrapolation) Algorithm 11.3B
114.C - Nonlinear Finite Difference Algorithm
11.4
114B.C -
Nonlinear Finite Difference (Richardson's Extrap.) Algorithm 11.4B
115.C - Piecewise Linear Rayleigh-Ritz Algorithm
11.5
116.C - Cubic Spline Rayleigh-Ritz Algorithm
11.6
CHAPTER
12
Numerical
Solutions
to
Partial-Differential
Equations
|
CHAPTER
12 Numerical Solutions to
Partial-Differential Equations
121.C - Poisson Equation Finite-Difference (Elliptic) Algorithm 12.1
122.C* -
Heat Equation Backward-Difference (Parabolic) Algorithm 12.2
122B.C* -
Heat Equation Forward-Difference (Parabolic) Algorithm 12.2B
122C.C* -
Heat Equation Richardson's Method (Parabolic) Algorithm 12.2C
123.C* -
Crank-Nicolson (Parabolic) Algorithm
12.3
124.C - Wave Equation Finite-Difference (Hyperbolic) Algorithm 12.4
125.C - Finite-Element Algorithm
12.5
126A.C -
Parabolic Equations With Newton Iteration in 1-D Algorithm 12.6A
127A.C -
Parabolic Equations With Newton Iteration in 2-D Algorithm 12.7A
128A.C -
Elliptic Equations With Newton Iteration in 2-D Algorithm 12.8A
129A.C -
Biharmonic Equation Using Gauss-Jordan Method Algorithm 12.9A
APPENDIX
A
"Numerical
Analysis
Algorithms
in C"
Supporting
C
Source Code
|
APPENDIX
A "Numerical Analysis Algorithms in
C" Supporting C Source Code
041EE.C* - Composite Simpson's Rule Using the
Equation Evaluator Routines
041FUN.C - Composite Simpson's Rule as a Function
CONVERT.C* - Converts Files from 8-bit Extended ASCII to
7-bit Standard ASCII
EE.C* - Command-Line Calculator using
the Equation Evaluator Routines