# NOTE:
# This implementation prioritizes clarity and correctness over optimization.
# Further performance improvements can be made if needed.
# (c) May 26, 2026 by James Pate Willims, Jr.
# I had some help from the Microsoft Copilot
# to calculate runtimes and define matrices
# Computes the potential in a rectangle
# Reference: "Boundary Value Problems
# Second Edition" by David L. Powers
# See pages 179 to 182 for the analytic
# solution of this Laplace Equation
# Stand alone application using
# Microsoft Visual Studio 2022
# Community Version
import math
import time
xi = yi = 10
u = [[0.0 for _ in range(xi + 2)] for _ in range(yi + 2)]
v = [[0.0 for _ in range(xi + 2)] for _ in range(yi + 2)]
def ComputeBoundaryValues(x, y):
if x == 0:
return 0
if x == 1:
return 0
if y == 0 or y == 1:
if x > 0.0 and x < 0.5:
return 2.0 * x
elif x >= 0.5 and x < 1.0:
return 2.0 - 2.0 * x
return 0.0
def ComputeParams(its, norm, params):
params['iterations'] = its
params['norm'] = norm
def Compute(h, k, xi, yi, maxIts, params):
# Use a simple fixed-point iteration to
# compute an approximate solution
for i in range(0, xi + 1):
for j in range(0, yi + 1):
u[i][j] = ComputeBoundaryValues(i * h, j * k)
for its in range(1, maxIts + 1):
for i in range(1, xi):
for j in range(1, yi):
u[i][j] = 0.25 * (u[i + 1][j] + u[i - 1][j] + u[i][j + 1] + u[i][j - 1]);
norm = 0
for i in range(0, xi + 1):
for j in range(0, yi + 1):
norm += math.fabs(u[i][j] * u[i][j])
norm = math.sqrt(norm)
params['iterations'] = its
params['norm'] = norm
def f(x, y):
# Analytic solution series expansion n = 1 to 100
sum = 0.0
for n in range(1, 101):
factor1 = math.sin(n * math.pi / 2.0) / (n * n)
factor2 = math.sinh(n * math.pi * y)
factor3 = math.sinh(n * math.pi * (1 - y))
factor4 = math.sin(n * math.pi * x)
term = (factor2 + factor3) / math.sinh(n * math.pi)
sum += factor1 * term * factor4
return 8.0 * sum / (math.pi * math.pi)
avgPE = 0
deltaX = 1.0 / xi
deltaY = 1.0 / yi
maxIts = 50
start_time = time.perf_counter()
for i in range(0, xi + 1):
for j in range(0, yi + 1):
v[i][j] = f(i * deltaX, j * deltaY)
minPE = +1000000000
maxPE = -1000000000
params = {}
Compute(deltaX, deltaY, xi, yi, maxIts, params)
print("Approximate\tAnalytic\tPercent Error")
for i in range(0, xi + 1):
for j in range(0, yi + 1):
if (math.fabs(u[i][j]) > 1.0e-12 and
math.fabs(v[i][j]) > 1.0e-12):
pe = 100.0 * math.fabs((v[i][j] - u[i][j]) / v[i][j])
else:
pe = 0.0
avgPE += pe
if (pe < minPE):
minPE = pe
if (pe > maxPE):
maxPE = pe
if math.fabs(pe) != 0.0:
print("{:10.8f}".format(u[i][j]), "\t", "{:10.8f}".format(v[i][j]), "\t", "{:10.8f}".format(pe))
avgPE /= (xi * yi)
end_time = time.perf_counter()
# Calculate elapsed time in milliseconds
elapsed_ms = (end_time - start_time) * 1000
print("Iterations = ", params['iterations'])
print("Frobenius Norm = ", params['norm'])
print("Minimum Percent Error = ", "{:10.8f}".format(minPE))
print("Average Percent Error = ", "{:10.8f}".format(avgPE))
print("Maximum Percent Error = ", "{:10.8f}".format(maxPE))
print("Elapsed Milliseconds = ", "{:10.8f}".format(elapsed_ms))
Approximate Analytic Percent Error
0.20000000 0.19999972 0.00013831
0.16633455 0.16663592 0.18085704
0.13739427 0.13768928 0.21425509
0.11591159 0.11605132 0.12040154
0.10292732 0.10291871 0.00836375
0.09864668 0.09854114 0.10710198
0.10305612 0.10291871 0.13351975
0.11613103 0.11605132 0.06868445
0.13763395 0.13768928 0.04018144
0.16650309 0.16663592 0.07971563
0.20000000 0.19999972 0.00013831
0.40000000 0.39999927 0.00018169
0.32813882 0.32854798 0.12453472
0.26763492 0.26776105 0.04710489
0.22369849 0.22344522 0.11334607
0.19755216 0.19705465 0.25246951
0.18899096 0.18834090 0.34515051
0.19778524 0.19705465 0.37075402
0.22409557 0.22344522 0.29105301
0.26806863 0.26776105 0.11486989
0.32844379 0.32854798 0.03171204
0.40000000 0.39999927 0.00018169
0.60000000 0.59999811 0.00031532
0.47888974 0.47875999 0.02710181
0.38176991 0.38059768 0.30799710
0.31425594 0.31267225 0.50650231
0.27518847 0.27354681 0.60013876
0.26255192 0.26082096 0.66365837
0.27549367 0.27354681 0.71170992
0.31477587 0.31267225 0.67278721
0.38233779 0.38059768 0.45720485
0.47928905 0.47875999 0.11050679
0.60000000 0.59999811 0.00031532
0.80000000 0.79999202 0.00099701
0.60602379 0.60222488 0.63081180
0.46685956 0.46199176 1.05365615
0.37704317 0.37308607 1.06063957
0.32711029 0.32392135 0.98448025
0.31121899 0.30818168 0.98555933
0.32745161 0.32392135 1.08985035
0.37762462 0.37308607 1.21648870
0.46749464 0.46199176 1.19112076
0.60647034 0.60222488 0.70496220
0.80000000 0.79999202 0.00099701
1.00000000 0.99594729 0.40692036
0.67874673 0.65811281 3.13531720
0.50319768 0.49282441 2.10486107
0.40066316 0.39470092 1.51057096
0.34574699 0.34157202 1.22228048
0.32848272 0.32468552 1.16950228
0.34608839 0.34157202 1.32223096
0.40124475 0.39470092 1.65792212
0.50383291 0.49282441 2.23375660
0.67919339 0.65811281 3.20318626
1.00000000 0.99594729 0.40692036
0.80000000 0.79999202 0.00099701
0.60615279 0.60222488 0.65223206
0.46709299 0.46199176 1.10418350
0.37734885 0.37308607 1.14257161
0.32745218 0.32392135 1.09002599
0.31156100 0.30818168 1.09653477
0.32776105 0.32392135 1.18538026
0.37787503 0.37308607 1.28360546
0.46766769 0.46199176 1.22857864
0.60655688 0.60222488 0.71933113
0.80000000 0.79999202 0.00099701
0.60000000 0.59999811 0.00031532
0.47910949 0.47875999 0.07300025
0.38216756 0.38059768 0.41247566
0.31477665 0.31267225 0.67303766
0.27577086 0.27354681 0.81304189
0.26313452 0.26082096 0.88702848
0.27602079 0.27354681 0.90440981
0.31520243 0.31267225 0.80920943
0.38263258 0.38059768 0.53465917
0.47943646 0.47875999 0.14129609
0.60000000 0.59999811 0.00031532
0.40000000 0.39999927 0.00018169
0.32837881 0.32854798 0.05149022
0.26806920 0.26776105 0.11508237
0.22426717 0.22344522 0.36785094
0.19818820 0.19705465 0.57524397
0.18962723 0.18834090 0.68297865
0.19836093 0.19705465 0.66290001
0.22456142 0.22344522 0.49953914
0.26839057 0.26776105 0.23510691
0.32860478 0.32854798 0.01728752
0.40000000 0.39999927 0.00018169
0.20000000 0.19999972 0.00013831
0.16650327 0.16663592 0.07960469
0.13769959 0.13768928 0.00748883
0.11631140 0.11605132 0.22411146
0.10337449 0.10291871 0.44285504
0.09909401 0.09854114 0.56105764
0.10346087 0.10291871 0.52678322
0.11645855 0.11605132 0.35090580
0.13786030 0.13768928 0.12420922
0.16661627 0.16663592 0.01179315
0.20000000 0.19999972 0.00013831
Iterations = 50
Frobenius Norm = 4.028216200275417
Minimum Percent Error = 0.00000000
Average Percent Error = 0.54286140
Maximum Percent Error = 3.20318626
Elapsed Milliseconds = 36.23520000
Press any key to continue . . .
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.
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Numerical Explorations 