CSC_349_HANDOUT#15

CSC_349_HANDOUT#15 - COMPUTER SCIENCE 349A Handout Number...

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1 COMPUTER SCIENCE 349A Handout Number 15 NAÏVE GAUSSIAN ELIMINATION (Section 9.2) Notation for a system of n linear equations in n unknowns: Ax = b , where A is an n × n nonsingular matrix and x and b are (column) vectors with n entries. Equivalently, n n n n n n n n n n b x a x a x a b x a x a x a b x a x a x a = + + + = + + + = + + + L M L L 2 2 1 1 2 2 2 22 1 21 1 1 2 12 1 11 Given data A and b , the problem is to determine x . Examples and motivation : see, for example, the Case Studies in Chapter 12 of the textbook. Theoretical basis for the Gaussian elimination algorithm . You can apply any of the following three elementary row operations to Ax = b : (i) multiply any equation i E by a nonzero constant λ (ii) replace equation i E by E i + E j (iii) interchange any two equations i E and j E in order to reduce A to upper triangular form . This triangular system is easily solved for x using back-substitution. The Gaussian elimination algorithm consists of two parts: (i) forward elimination -- the reduction of the coefficient matrix A to upper triangular form (ii) back-substitution -- solving this "reduced" upper triangular system for x .
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2 Numeric Example with n = 3. The given linear system is x 1 + x 2 x 3 =− 2 2 x 1 x 2 + 3 x 3 = 14 or x 1 2 x 2 + x 3 = 3 ⎡− = 3
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CSC_349_HANDOUT#15 - COMPUTER SCIENCE 349A Handout Number...

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