CSC201 Solution of Simultaneous Algebraic Equations Using Lower-Upper Decomposition Method
Solved assignment explaining LU decomposition for solving simultaneous algebraic equations.
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CSC201 Solution of Simultaneous Algebraic Equations Using Lower-
Upper Decomposition Method
CSC201 Project #3 Summer15
Print Name: _____________________________________________________
Academic Honesty Policy: Students at the college are expected to be honest and forthright in
their academic endeavors. To falsify the results of one’s research, to steal the words or ideas
of another, to cheat on an examination, or to allow another to commit an act of academic
dishonesty corrupts the essential process by which knowledge is advanced. It is the official
policy of the Northern Virginia Community College that all acts or attempted acts of alleged
academic dishonesty be reported to the Dean of Students Office.
By signing below, I acknowledge that I have read the above and that I have neither given nor
received assistance on this examination.
Sign Name: _______________________________________________________
3. Solution of Simultaneous Algebraic Equations ***:
In algebra, we often have to solve simultaneous differential equations. If the equations
are linear and independent, there is a unique solution when the number of equations
equals the number of variables. If there are only two variables and two equations, it’s
easy, but as the number of equations and unknowns increases, the problem becomes
more difficult. Imagine, for example trying to solve 100 equations in 100 unknowns.
That's an unthinkably difficult manual exercise, but a good computer program can do
it in less than a second.
Suppose you want the solution to this pair of equations:
2 * x + y = 1
4 * y = 12
In the first equation, the coefficient of x is 2 and the coefficient of y is 1. In the second
equation, the coefficient of x is 0, and the coefficient of y is 4. The right-side values are
1 and 12, respectively. It's easy to solve this pair of equations by hand. The second
equation says y = 12/4 = 3, and substituting this back into the first equation gives x =
(1-3)/2 = -1. The program you'll write will solve this problem like this:
Sample session:
Number of Equations & Unknowns: 2
Enter equation 0's coefficients separated by spaces:
Upper Decomposition Method
CSC201 Project #3 Summer15
Print Name: _____________________________________________________
Academic Honesty Policy: Students at the college are expected to be honest and forthright in
their academic endeavors. To falsify the results of one’s research, to steal the words or ideas
of another, to cheat on an examination, or to allow another to commit an act of academic
dishonesty corrupts the essential process by which knowledge is advanced. It is the official
policy of the Northern Virginia Community College that all acts or attempted acts of alleged
academic dishonesty be reported to the Dean of Students Office.
By signing below, I acknowledge that I have read the above and that I have neither given nor
received assistance on this examination.
Sign Name: _______________________________________________________
3. Solution of Simultaneous Algebraic Equations ***:
In algebra, we often have to solve simultaneous differential equations. If the equations
are linear and independent, there is a unique solution when the number of equations
equals the number of variables. If there are only two variables and two equations, it’s
easy, but as the number of equations and unknowns increases, the problem becomes
more difficult. Imagine, for example trying to solve 100 equations in 100 unknowns.
That's an unthinkably difficult manual exercise, but a good computer program can do
it in less than a second.
Suppose you want the solution to this pair of equations:
2 * x + y = 1
4 * y = 12
In the first equation, the coefficient of x is 2 and the coefficient of y is 1. In the second
equation, the coefficient of x is 0, and the coefficient of y is 4. The right-side values are
1 and 12, respectively. It's easy to solve this pair of equations by hand. The second
equation says y = 12/4 = 3, and substituting this back into the first equation gives x =
(1-3)/2 = -1. The program you'll write will solve this problem like this:
Sample session:
Number of Equations & Unknowns: 2
Enter equation 0's coefficients separated by spaces:
2 1
Enter right-side value: 1
Enter equation 1's coefficients separated by spaces:
0 4
Enter right-side value: 12
Equations
2.0 1.0 1.0
0.0 4.0 12.0
Solution
-1.0
3.0
Verification
1.0 1.0
12.0 12.0
The numbers under “Verification” show a comparison between the original right-side
values (first column) with what the computer gets when it substitutes it’s solution back
into the original equations (second column). It’s not worthwhile writing a computer
program to solve a simple problem like this, but it would be worthwhile if you had to
solve 10 equations with 10 unknowns, and suppose you had 200 equations with 200
unknowns? This project’s program will be able to solve even a problem of that size,
with no sweat.
Often it’s useful to put a hard-coded default example into your code, to keep you from
having to re-enter everything when debugging. You could use an unreasonable input
like zero to generate a test case:
Another sample session:
Number of Equations & Unknowns: 0
Equations
-6.0 33.0 16.0 -36.0
-7.0 34.0 -8.0 43.0
-25.0 22.0 9.0 -46.0
Solution
1.310701412279147
0.7545943390651776
-3.3148377947172487
Verification
-36.0 -36.0
Enter right-side value: 1
Enter equation 1's coefficients separated by spaces:
0 4
Enter right-side value: 12
Equations
2.0 1.0 1.0
0.0 4.0 12.0
Solution
-1.0
3.0
Verification
1.0 1.0
12.0 12.0
The numbers under “Verification” show a comparison between the original right-side
values (first column) with what the computer gets when it substitutes it’s solution back
into the original equations (second column). It’s not worthwhile writing a computer
program to solve a simple problem like this, but it would be worthwhile if you had to
solve 10 equations with 10 unknowns, and suppose you had 200 equations with 200
unknowns? This project’s program will be able to solve even a problem of that size,
with no sweat.
Often it’s useful to put a hard-coded default example into your code, to keep you from
having to re-enter everything when debugging. You could use an unreasonable input
like zero to generate a test case:
Another sample session:
Number of Equations & Unknowns: 0
Equations
-6.0 33.0 16.0 -36.0
-7.0 34.0 -8.0 43.0
-25.0 22.0 9.0 -46.0
Solution
1.310701412279147
0.7545943390651776
-3.3148377947172487
Verification
-36.0 -36.0
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Document Details
University
Obafemi Awolowo University
Subject
Mathematics