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Method If an nxn matrix A has an LU-factorization, then the solution of AX = b can be determined by a Forward substitution followed by a Back substitution [A] = [L][U] where [L] = lower triangular matrix all of whose diagonal entries are different from zero. [U] = upper triangular matrix all of whose diagonal entries are different from zero. LU Decomposition

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How can this be used? Given [A][X] = [b] 1.Decompose [A] into [L] and [U] (LU)X = b L(UX) = b ; Let UX = z then LZ = b 1.Use Forward substitution to Solve [L][Z] = [b] for [Z] 2.Use Back substitution to Solve [U][X] = [Z] for [X]

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Exercise 2.5 ; Page # 136; Qn # 1 3

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Ax = b (LU)X = b L(UX) = b LZ = b; Let UX = Z 4

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[L][Z] = [b] Solve for [Z] using Forward Substitution :

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Complete the forward substitution to solve for [Z]

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[U][X] = [Z] Solve for [X] using Back Substitution :

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The Solution is :

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Method: [A] Decompose to [L] and [U] Storage of Multipliers Scheme :

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Exercise 2.5; Page #136; Qn # 5 Finding the [U] matrix Step 1:

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Finding the [U] Matrix Step 2: Matrix after Step 1:

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Finding the [L] matrix http://numericalmethods.eng.usf.edu

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Finding the [L] Matrix

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[L][Z] = [b] Solve for [Z] using Forward Substitution :

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Complete the forward substitution to solve for [Z]

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[U][X] = [Z] Solve for [X] using Back Substitution :

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The Solution is :

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Limitations to use this Procedure : Interchanging of any two rows not allowed. Only elementary row operation permitted is the one that subtract a multiple of one row to another. In matrix A, if then this procedure fails. In matrix if, or in if then this procedure fails.

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Test for you? Exercise 2.5 ; Page # 136; Qn # 7 19

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Arithmetic Operations on Matrices. 1. Definition of Matrix 2. Column, Row and Square Matrix 3. Addition and Subtraction of Matrices 4. Multiplying Row.

Arithmetic Operations on Matrices. 1. Definition of Matrix 2. Column, Row and Square Matrix 3. Addition and Subtraction of Matrices 4. Multiplying Row.

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