SE301: Numerical Methods Topic 3: Solution of Systems of Linear Equations Lectures 12-17:

CISE301_Topic3KFUPM * SE301: Numerical Methods Topic 3: Solution of Systems of Linear Equations Lectures 12-17: KFUPM

Read Chapter 9 of the textbook

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CISE301_Topic3KFUPM *Lecture 12Vector, Matrices, andLinear Equations

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CISE301_Topic3KFUPM *VECTORS

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CISE301_Topic3KFUPM *MATRICES

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CISE301_Topic3KFUPM *Determinant of a MATRICES

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CISE301_Topic3KFUPM *Adding and Multiplying Matrices

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CISE301_Topic3KFUPM *Systems of Linear Equations

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CISE301_Topic3KFUPM *Solutions of Linear Equations

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CISE301_Topic3KFUPM *Solutions of Linear EquationsA set of equations is inconsistent if there exists no solution to the system of equations:

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CISE301_Topic3KFUPM *Solutions of Linear EquationsSome systems of equations may have infinite number of solutions

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CISE301_Topic3KFUPM *Graphical Solution of Systems ofLinear Equationssolution

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CISE301_Topic3KFUPM *Cramers Rule is Not Practical

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CISE301_Topic3KFUPM * Naive Gaussian Elimination Examples

Lecture 13 Naive Gaussian Elimination

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CISE301_Topic3KFUPM *Naive Gaussian EliminationThe method consists of two steps:Forward Elimination: the system is reduced to upper triangular form. A sequence of elementary operations is used. Backward Substitution: Solve the system starting from the last variable.

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CISE301_Topic3KFUPM *Elementary Row Operations

Adding a multiple of one row to another

Multiply any row by a non-zero constant

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CISE301_Topic3KFUPM *ExampleForward Elimination

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CISE301_Topic3KFUPM *ExampleBackward Substitution

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CISE301_Topic3KFUPM *Forward Elimination

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CISE301_Topic3KFUPM *Backward Substitution

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CISE301_Topic3KFUPM * Summary of the Naive Gaussian Elimination Example How to check a solution Problems with Naive Gaussian Elimination Failure due to zero pivot element ErrorLecture 14Naive Gaussian Elimination

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CISE301_Topic3KFUPM *Naive Gaussian EliminationThe method consists of two stepsForward Elimination: the system is reduced to upper triangular form. A sequence of elementary operations is used.

Backward Substitution: Solve the system starting from the last variable. Solve for xn ,xn-1,x1.

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CISE301_Topic3KFUPM *Example 1

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CISE301_Topic3KFUPM *Example 1Backward Substitution

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CISE301_Topic3KFUPM *How Do We Know If a Solution is Good or NotGiven AX=BX is a solution if AX-B=0Due to computation error AX-B may not be zeroCompute the residuals R=|AX-B|One possible test is ?????

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CISE301_Topic3KFUPM *Determinant

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CISE301_Topic3KFUPM *How Many Solutions Does a System of Equations AX=B Have?

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CISE301_Topic3KFUPM *Examples

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CISE301_Topic3KFUPM *Lectures 15-16:Gaussian Elimination with Scaled Partial Pivoting Problems with Naive Gaussian Elimination Definitions and Initial step Forward Elimination Backward substitution Example

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CISE301_Topic3KFUPM *Problems with Naive Gaussian EliminationThe Naive Gaussian Elimination may fail for very simple cases. (The pivoting element is zero).

Very small pivoting element may result in serious computation errors

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CISE301_Topic3KFUPM *Example 2Initialization stepScale vector:disregard sign find largest in magnitude in each row

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CISE301_Topic3KFUPM *Why Index Vector?Index vectors are used because it is much easier to exchange a single index element compared to exchanging the values of a complete row.In practical problems with very large N, exchanging the contents of rows may not be practical since they could be stored at different locations.

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CISE301_Topic3KFUPM *Example 2Forward Elimination-- Step 1: eliminate x1

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CISE301_Topic3KFUPM *Example 2Forward Elimination-- Step 1: eliminate x1First pivot equation

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CISE301_Topic3KFUPM *Example 2Forward Elimination-- Step 3: eliminate x3Third pivot equation

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CISE301_Topic3KFUPM *Example 3Initialization step

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CISE301_Topic3KFUPM *How Good is the Solution?

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CISE301_Topic3KFUPM *Remarks: We use index vector to avoid the need to move the rows which may not be practical for large problems.If we order the equation as in the last value of the index vector, we have a triangular form. Scale vector is formed by taking maximum in magnitude in each row. Scale vector do not change.The original matrices A and B are used in checking the residuals.

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CISE301_Topic3KFUPM *Lecture 17 Tridiagonal & Banded Systems and Gauss-Jordan Method Tridiagonal Systems Diagonal Dominance Tridiagonal Algorithm Examples Gauss-Jordan Algorithm

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CISE301_Topic3KFUPM *Tridiagonal Systems: The non-zero elements are in the main diagonal, super diagonal and subdiagonal.aij=0 if |i-j| > 1

Tridiagonal Systems

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CISE301_Topic3KFUPM *Occur in many applicationsNeeds less storage (4n-2 compared to n2 +n for the general cases)Selection of pivoting rows is unnecessary (under some conditions)Efficiently solved by Gaussian elimination

Tridiagonal Systems

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CISE301_Topic3KFUPM *Based on Naive Gaussian elimination. As in previous Gaussian elimination algorithmsForward elimination stepBackward substitution stepElements in the super diagonal are not affected. Elements in the main diagonal, and B need updating

Algorithm to Solve Tridiagonal Systems

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CISE301_Topic3KFUPM *Tridiagonal System

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CISE301_Topic3KFUPM *Diagonal Dominance

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CISE301_Topic3KFUPM *Diagonally Dominant Tridiagonal SystemA tridiagonal system is diagonally dominant if

Forward Elimination preserves diagonal dominance

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CISE301_Topic3KFUPM *Solving Tridiagonal System

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CISE301_Topic3KFUPM *Example

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CISE301_Topic3KFUPM *ExampleBackward SubstitutionAfter the Forward Elimination:

Backward Substitution:

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CISE301_Topic3KFUPM *Gauss-Jordan MethodThe method reduces the general system of equations AX=B to IX=B where I is an identity matrix.

Only Forward elimination is done and no substitution is needed.

It has the same problems as Naive Gaussian elimination and can be modified to do partial scaled pivoting.

It takes 50% more time than Naive Gaussian method.

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CISE301_Topic3KFUPM *Gauss-Jordan MethodExample

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CISE301_Topic3KFUPM *Gauss-Jordan MethodExample

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SE301: Numerical Methods Topic 3: Solution of Systems of Linear Equations Lectures 12-17: