lecture 7 - systems of equations cven 302 june 17, 2002
TRANSCRIPT
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Lecture 7 - Systems of EquationsLecture 7 - Systems of Equations
CVEN 302
June 17, 2002
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Lecture’s GoalsLecture’s Goals
• Discuss how to solve systems– Gaussian Elimination– Gaussian Elimination with Pivoting– Tridiagonal Solver
• Problems with the technique
• Examples
• Iterative Techniques
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Computer ProgramComputer Program
The program GEdemo(A,b) does the Gaussian
elimination for a square matrix (nxn). It does
not do any pivoting and works for only one
{b} vector.
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Test the ProgramTest the Program
• Example 1
• Example 2
• New Matrix
2X1 + 4X2 - 2 X3 - 2 X4 = - 4
1X1 + 2X2 + 4X3 - 3 X4 = 5
- 3X1 - 3X2 + 8X3 - 2X4 = 7
- X1 + X2 + 6X3 - 3X4 = 7
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Problem with Gaussian Problem with Gaussian EliminationElimination
• The problem can occur when a zero appears in the diagonal and makes a simple Gaussian elimination impossible.
• Pivoting changes the matrix so that it will become diagonally dominate and reduce the round-off and truncation errors in the solving the matrix.
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Example of PivotingExample of Pivoting
2 X1 + 4 X2 - 2 X3 = 10
X1 + 2 X2 + 4 X3 = 6
2 X1 + 2 X2 + 1X3 = 2
Answer [X1 X2 X3 ] = [-3.40, 4.30, 0.20 ]
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Computer ProgramComputer Program
• GEPivotdemo(A,b) is a program, which will do a Gaussian elimination on matrix A with pivoting technique to make matrix diagonally dominate.
• The program is modification to handle a single value of {b}
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Question?Question?
• How would you modify the programs to handle multiple inputs?
• What is diagonal matrix, upper triangular matrix, and lower triangular matrix?
• Can you do a column exchange and how would you handle the problem if it works?
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Gaussian EliminationGaussian Elimination
• If the diagonal is not dominate the problem can have round off error and truncation errors.
• The scaling will result in problems
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Question?Question?
• What happens with the following example?
0.0001X1 + 0.5 X2 = 0.5
0.4000X1 - 0.3 X2 = 0.1
• What happens is the second equation becomes: 0.4000X1 - 2000 X2 = -2000
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Question?Question?
• What happens with the following example for values with two-significant figures?
0.4000 X1 - 0.3 X2 = 0.1
0.0001 X1 + 0.5 X2 = 0.5
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ScalingScaling
• Scaling is an operation of adjusting the coefficients of a set of equations so that they are all of the same magnitude.
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ScalingScaling
• A set of equations may involve relationships between quantities measured in a widely different units (N vs. kN, sec vs hrs, etc.) This may result in equation having very large number and others with very small , if we select pivoting may put numbers on the diagonal that are not large in comparison to other rows and create round-off errors that pivoting was suppose to avoid.
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ScalingScaling
• What happens with the following example?
3X1 + 2 X2 +100X3 = 105
- X1 + 3 X2 +100X3 = 102
X1 + 2 X2 - 1X3 = 2
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ScalingScaling
• The best way to handle the problem is to normalize the results.
0.03X1 + 0.02 X2 +1.00X3 = 1.05
- 0.01X1 + 0.03 X2 +1.00X3 = 1.02
0.50X1 +1.00 X2 - 0.50X3 = 1.00
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Gauss-Jordan MethodGauss-Jordan Method
• The Gauss-Jordan Method is similar to the Gaussian Elimination.
• The method requires almost 50% more operations.
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Gauss-Jordan MethodGauss-Jordan Method
The Gauss-Jordan method changes the matrix into the identity matrix.
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Gauss-Jordan MethodGauss-Jordan Method
There are one phases to the solving technique
• Elimination --- use row operations to convert the matrix into an identity matrix.
• The new b vector is the solution to the x values.
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Gauss-Jordan AlgorithmGauss-Jordan Algorithm
[A]{x} ={b}
• Augment the n x n coefficient matrix with the vector of right hand sides to form a n x (n+1)
• Interchange rows if necessary to make the value a11 with the largest magnitude of any coefficient in the first row
• Create zero in 2nd through nth row in first row by subtracting ai1 / a11 times first row from ith row
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Gauss-Jordan Elimination Gauss-Jordan Elimination AlgorithmAlgorithm
• Repeat (2) & (3) for first through the nth rows, putting the largest magnitude coefficient in the diagonal by interchanging rows (consider only row j to n ) and then subtract times the jth row from the ith row so as to create zeros in all positions of jth column and the diagonal becomes all ones
• Solve for all of the equations, xi = ai,n+1
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Example 1Example 1
X1 + 3X2 = 5
2X1 + 4X2 = 6
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Example 2Example 2
-3X1 + 2X2 - X3 = -1
6X1 - 6X2 + 7X3 = -7
3X1 - 4X2 + 4X3 = -6
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Band SolverBand Solver
• Large matrices tend to be banded, which means that the matrix has a band of non-zero coefficients and zeroes on the outside of the matrix.
• The simplest of the methods is the Thomas Method, which is used for a tridiagonal matrix.
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Advantages of Band SolversAdvantages of Band Solvers
• The method reduce the number of operations and save the matrix in smaller amount of memory.
• The band solver is faster and is useful for large scale matrices.
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Thomas MethodThomas Method
• The method takes advantage of the bandedness of the matrix.
• The technique uses a two phase process.
– The first phase is to obtain the coefficients from the sweep.
– The second phase solves for the x values.
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Thomas MethodThomas Method
• The first phase starts with the first row of coefficients scales the a and r coefficients.
• The second phase solves for x values using the a and r coefficients.
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Thomas MethodThomas Method
• The program for the method is given as demoThomas(a,d,b,r)
• The algorithm is from the textbook, where a,d,b, & r are vectors from the matrix.
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Iterative TechniquesIterative Techniques
• The method of solving simultaneous linear algebraic equations using Gaussian Elimination and the Gauss-Jordan Method. These techniques are known as direct methods. Problems can arise from round-off errors and zero on the diagonal.
• One means of obtaining an approximate solution to the equations is to use an “educated guess”.
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Iterative MethodsIterative Methods
• We will look at three iterative methods:– Jacobi Method– Gauss-Seidel Method– Successive over Relaxation (SOR)
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Convergence RestrictionsConvergence Restrictions
• There are two conditions for the iterative method to converge.
– Necessary that 1 coefficient in each equation is dominate.
– The sufficient condition is that the diagonal is dominate.
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Jacobi IterationJacobi Iteration
• If the diagonal is dominant, the matrix can be rewritten in the following form
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Jacobi IterationJacobi Iteration
• The technique can be rewritten in a shorthand fashion, where D is the diagonal, A” is the matrix without the diagonal and c is the right-hand side of the equations.
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SummarySummary
• Scaling of the problem will help in the convergence.
• Gauss-Jordan method is more computational intense and does not improve the round-off errors. However, it is useful for finding matrix inverses.
• Banded matrix solvers are faster and use less memory.
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HomeworkHomework
• Check the Homework webpage