lecture8_v2 - CMPSC/MATH 451 Numerical Computations Lecture...

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CMPSC/MATH 451 Numerical Computations Lecture 8 September 9, 2011 Prof. Kamesh Madduri
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Review of last class Transformations Triangular systems Forward substitution Back substitution 2
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Gaussian Elimination Covered on blackboard, corresponding slides from textbook follow. Also see additional notes for examples covered in class. 3
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Existence, Uniqueness, and Conditioning Solving Linear Systems Special Types of Linear Systems Software for Linear Systems Triangular Systems Gaussian Elimination Updating Solutions Improving Accuracy Example: Triangular Linear System 2 4 - 2 0 1 1 0 0 4 x 1 x 2 x 3 = 2 4 8 Using back-substitution for this upper triangular system, last equation, 4 x 3 = 8 , is solved directly to obtain x 3 = 2 Next, x 3 is substituted into second equation to obtain x 2 = 2 Finally, both x 3 and x 2 are substituted into first equation to obtain x 1 = - 1 Michael T. Heath Scientific Computing 33 / 88
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Existence, Uniqueness, and Conditioning Solving Linear Systems Special Types of Linear Systems Software for Linear Systems Triangular Systems Gaussian Elimination Updating Solutions Improving Accuracy Elimination To transform general linear system into triangular form, we need to replace selected nonzero entries of matrix by zeros This can be accomplished by taking linear combinations of rows Consider 2-vector a = a 1 a 2 If a 1 6 = 0 , then 1 0 - a 2 /a 1 1 a 1 a 2 = a 1 0 Michael T. Heath Scientific Computing 34 / 88
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Existence, Uniqueness, and Conditioning Solving Linear Systems Special Types of Linear Systems Software for Linear Systems Triangular Systems Gaussian Elimination Updating Solutions Improving Accuracy Elementary Elimination Matrices More generally, we can annihilate all entries below k th position in n -vector a by transformation M k a = 1 · · · 0 0 · · · 0 . . . . . . . . . . . . . . . . . . 0 · · · 1 0 · · · 0 0 · · · - m k +1 1 · · · 0 . . . . . . . . . . . . . . . . . . 0 · · · - m n 0 · · · 1 a 1 . . . a k a k +1 . . . a n = a 1 .
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