Double Integration of Six Functions Using Three Different Quadrature Algorithms (c) January 15, 2024, by James Pate Williams, Jr. All Applicable Rights Reserved

I utilized six functions that have two independent variables. The integration methods were a Monte Carlo probabilistic procedure, TRICUB from “A Numerical Library in C for Scientists and Engineers” by H. T. Lau, PhD, and an algorithm from the ACM publication. I translated the ACM method from FORTRAN to C. See the following online references:

https://math.libretexts.org/Bookshelves/Calculus/Vector_Calculus_(Corral)/03%3A_Multiple_Integrals/3.04%3A_Numerical_Approximation_of_Multiple_Integrals
https://tutorial.math.lamar.edu/Classes/CalcIII/IteratedIntegrals.aspx

== Menu ==

1 8 * x + 6 * y

2 6 * x * y * y

3 2 * x - 4 * y * y * y

4 x * x * y * y + cos(pi * x) + sin(pi * y)

5 pow(2 * x + 3 * y, -2)

6 x * exp(x * y)

7 Exit

Option: 1

f(x, y) = 8x + 6y

[0.000000, 1.000000] x [0.000000, 2.000000]

N       Integral        +-Error         % Error

10      18.398218       2.687822        8.008911

100     19.546981       0.838888        2.265096

1000    19.859442       0.266101        0.702789

10000   20.152819       0.082772        0.764093

100000  20.014598       0.026273        0.072988

1000000 20.013430       0.008323        0.067150

NUMAL Integral Value and Percent Error: 22.666667       13.333333

nQuadrature Integral Value and Percent Error: 19.999968 0.000160

== Menu ==

1 8 * x + 6 * y

2 6 * x * y * y

3 2 * x - 4 * y * y * y

4 x * x * y * y + cos(pi * x) + sin(pi * y)

5 pow(2 * x + 3 * y, -2)

6 x * exp(x * y)

7 Exit

Option:

== Menu ==

1 8 * x + 6 * y

2 6 * x * y * y

3 2 * x - 4 * y * y * y

4 x * x * y * y + cos(pi * x) + sin(pi * y)

5 pow(2 * x + 3 * y, -2)

6 x * exp(x * y)

7 Exit

Option: 2

f(x, y) = 6 * x * y * y

[1.000000, 2.000000] x [2.000000, 4.000000]

N       Integral        +-Error         % Error

10      6.400786        2.133333        92.380017

100     7.528110        0.951714        91.037965

1000    7.951653        0.294436        90.533746

10000   8.083377        0.094959        90.376933

100000  8.001852        0.029859        90.473986

1000000 8.010457        0.009474        90.463742

NUMAL Integral Value and Percent Error: 204.800000      143.809524

nQuadrature Integral Value and Percent Error: 167.999730        99.999679

== Menu ==

1 8 * x + 6 * y

2 6 * x * y * y

3 2 * x - 4 * y * y * y

4 x * x * y * y + cos(pi * x) + sin(pi * y)

5 pow(2 * x + 3 * y, -2)

6 x * exp(x * y)

7 Exit

Option:

== Menu ==

1 8 * x + 6 * y

2 6 * x * y * y

3 2 * x - 4 * y * y * y

4 x * x * y * y + cos(pi * x) + sin(pi * y)

5 pow(2 * x + 3 * y, -2)

6 x * exp(x * y)

7 Exit

Option: 3

f(x, y) = 2 * x - 4 * y * y * y

[-5.000000, 4.000000] x [0.000000, 3.000000]

N       Integral        +-Error         % Error

10      -645.214459     239.379239      14.654172

100     -436.170436     84.138767       42.305498

1000    -471.478924     25.861533       37.635063

10000   -486.066536     8.338088        35.705485

100000  -484.301805     2.644031        35.938915

1000000 -486.527596     0.839338        35.644498

NUMAL Integral Value and Percent Error: -765.000000     1.190476

nQuadrature Integral Value and Percent Error: -755.998734       0.000167

== Menu ==

1 8 * x + 6 * y

2 6 * x * y * y

3 2 * x - 4 * y * y * y

4 x * x * y * y + cos(pi * x) + sin(pi * y)

5 pow(2 * x + 3 * y, -2)

6 x * exp(x * y)

7 Exit

Option:

== Menu ==

1 8 * x + 6 * y

2 6 * x * y * y

3 2 * x - 4 * y * y * y

4 x * x * y * y + cos(pi * x) + sin(pi * y)

5 pow(2 * x + 3 * y, -2)

6 x * exp(x * y)

7 Exit

Option: 4

f(x, y) = x * x * y * y + cos(pi * x) + sin(pi * y)

[-2.000000, -1.000000] x [0.000000, 1.000000]

N       Integral        +-Error         % Error

10      1.103184        0.173977        22.003269

100     0.708005        0.063869        49.942995

1000    0.757961        0.021546        46.411062

10000   0.744889        0.006866        47.335224

100000  0.748463        0.002171        47.082576

1000000 0.746892        0.000686        47.193609

NUMAL Integral Value and Percent Error: 1.449491        2.481177

nQuadrature Integral Value and Percent Error: 1.414395  0.000160

== Menu ==

1 8 * x + 6 * y

2 6 * x * y * y

3 2 * x - 4 * y * y * y

4 x * x * y * y + cos(pi * x) + sin(pi * y)

5 pow(2 * x + 3 * y, -2)

6 x * exp(x * y)

7 Exit

Option:

== Menu ==

1 8 * x + 6 * y

2 6 * x * y * y

3 2 * x - 4 * y * y * y

4 x * x * y * y + cos(pi * x) + sin(pi * y)

5 pow(2 * x + 3 * y, -2)

6 x * exp(x * y)

7 Exit

Option: 5

f(x, y) = pow(2 * x + 3 * y, -2)

[0.000000, 1.000000] x [1.000000, 2.000000]

N       Integral        +-Error         % Error

10      0.555876        0.235032        1394.669315

100     0.752953        0.323309        1924.579264

1000    0.826769        0.275474        2123.058629

10000   0.927229        0.177631        2393.180482

100000  0.995193        0.097358        2575.927399

1000000 1.291253        0.127325        3371.987619

NUMAL Integral Value and Percent Error: 0.034125        8.243058

nQuadrature Integral Value and Percent Error: 0.037190  0.000639

== Menu ==

1 8 * x + 6 * y

2 6 * x * y * y

3 2 * x - 4 * y * y * y

4 x * x * y * y + cos(pi * x) + sin(pi * y)

5 pow(2 * x + 3 * y, -2)

6 x * exp(x * y)

7 Exit

Option:

== Menu ==

1 8 * x + 6 * y

2 6 * x * y * y

3 2 * x - 4 * y * y * y

4 x * x * y * y + cos(pi * x) + sin(pi * y)

5 pow(2 * x + 3 * y, -2)

6 x * exp(x * y)

7 Exit

Option: 6

f(x, y) = x * exp(x * y)

[0.000000, 1.000000] x [-1.000000, 2.000000]

N       Integral        +-Error         % Error

10      3.217934        1.437426        inf

100     5.562365        0.873166        inf

1000    5.325378        0.225123        inf

10000   5.497069        0.075770        inf

100000  5.362919        0.023344        inf

1000000 5.371802        0.007410        inf

NUMAL Integral Value and Percent Error: 6.162907        inf

nQuadrature Integral Value and Percent Error: 2.562339  inf

Unknown option, please try again.== Menu ==

1 8 * x + 6 * y

2 6 * x * y * y

3 2 * x - 4 * y * y * y

4 x * x * y * y + cos(pi * x) + sin(pi * y)

5 pow(2 * x + 3 * y, -2)

6 x * exp(x * y)

7 Exit

Option:
Unknown's avatar

Author: jamespatewilliamsjr

My whole legal name is James Pate Williams, Jr. I was born in LaGrange, Georgia approximately 70 years ago. I barely graduated from LaGrange High School with low marks in June 1971. Later in June 1979, I graduated from LaGrange College with a Bachelor of Arts in Chemistry with a little over a 3 out 4 Grade Point Average (GPA). In the Spring Quarter of 1978, I taught myself how to program a Texas Instruments desktop programmable calculator and in the Summer Quarter of 1978 I taught myself Dayton BASIC (Beginner's All-purpose Symbolic Instruction Code) on LaGrange College's Data General Eclipse minicomputer. I took courses in BASIC in the Fall Quarter of 1978 and FORTRAN IV (Formula Translator IV) in the Winter Quarter of 1979. Professor Kenneth Cooper, a genius poly-scientist taught me a course in the Intel 8085 microprocessor architecture and assembly and machine language. We would hand assemble our programs and insert the resulting machine code into our crude wooden box computer which was designed and built by Professor Cooper. From 1990 to 1994 I earned a Bachelor of Science in Computer Science from LaGrange College. I had a 4 out of 4 GPA in the period 1990 to 1994. I took courses in C, COBOL, and Pascal during my BS work. After graduating from LaGrange College a second time in May 1994, I taught myself C++. In December 1995, I started using the Internet and taught myself client-server programming. I created a website in 1997 which had C and C# implementations of algorithms from the "Handbook of Applied Cryptography" by Alfred J. Menezes, et. al., and some other cryptography and number theory textbooks and treatises.

Leave a comment