Blog Entry (c) Tuesday, July 23, 2024, by James Pate Williams, Jr. Mueller’s Method for Finding the Complex and/or Real Roots of a Complex and/or Real Polynomial

I originally implemented this algorithm in FORTRAN IV in the Summer Quarter of 1982 at the Georgia Institute of Technology. I was taking a course named “Scientific Computing I” taught by Professor Gunter Meyer. I made a B in the class. Later in 2015 I re-implemented the recipe in C# using Visual Studio 2008 Professional. VS 2015 did not have support for complex numbers nor large integers. In December of 2015 I upgraded to Visual Studio 2015 Professional which has support for big integers and complex numbers. I used Visual Studio 2019 Community version for this project. Root below should be function.

Degree (0 to quit) = 2
coefficient[2].real = 1
coefficient[2].imag = 0
coefficient[1].real = 1
coefficient[1].imag = 0
coefficient[0].real = 1
coefficient[0].imag = 0

zero[0].real = -5.0000000000e-01        zero[0].imag =  8.6602540378e-01
zero[1].real = -5.0000000000e-01        zero[1].imag = -8.6602540378e-01

root[0].real =  0.0000000000e+00        root[0].imag = -2.2204460493e-16
root[1].real =  3.3306690739e-16        root[1].imag = -7.7715611724e-16

Degree (0 to quit) = 3
coefficient[3].real = 1
coefficient[3].imag = 0
coefficient[2].real = 0
coefficient[2].imag = 0
coefficient[1].real = -18.1
coefficient[1].imag = 0
coefficient[0].real = -34.8
coefficient[0].imag = 0

zero[0].real = -2.5026325486e+00        zero[0].imag = -8.3036679880e-01
zero[1].real = -2.5026325486e+00        zero[1].imag =  8.3036679880e-01
zero[2].real =  5.0052650973e+00        zero[2].imag =  2.7417672687e-15

root[0].real =  0.0000000000e+00        root[0].imag =  1.7763568394e-15
root[1].real =  3.5527136788e-14        root[1].imag = -1.7763568394e-14
root[2].real =  2.8421709430e-14        root[2].imag =  1.5643985575e-13

Degree (0 to quit) = 5
coefficient[5].real = 1
coefficient[5].imag = 0
coefficient[4].real = 2
coefficient[4].imag = 0
coefficient[3].real = 3
coefficient[3].imag = 0
coefficient[2].real = 4
coefficient[2].imag = 0
coefficient[1].real = 5
coefficient[1].imag = 0
coefficient[0].real = 6
coefficient[0].imag = 0

zero[0].real = -8.0578646939e-01        zero[0].imag =  1.2229047134e+00
zero[1].real = -8.0578646939e-01        zero[1].imag = -1.2229047134e+00
zero[2].real =  5.5168546346e-01        zero[2].imag =  1.2533488603e+00
zero[3].real =  5.5168546346e-01        zero[3].imag = -1.2533488603e+00
zero[4].real = -1.4917979881e+00        zero[4].imag =  1.8329656063e-15

root[0].real =  8.8817841970e-16        root[0].imag =  4.4408920985e-16
root[1].real = -2.6645352591e-15        root[1].imag = -4.4408920985e-16
root[2].real =  8.8817841970e-16        root[2].imag =  1.7763568394e-15
root[3].real =  3.4638958368e-14        root[3].imag = -1.4210854715e-14
root[4].real =  8.8817841970e-16        root[4].imag =  2.0710031449e-14

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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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