linear_systems - Chapter 2 Linear Systems of Equations...

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Chapter 2 Linear Systems of Equations Existence, uniqueness Conditioning (sensitivity of solution) • If A is nonsingular, inverse A -1 exists x=A -1 b
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Gaussian Elimination with Partial Pivoting 1 3 2 7 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ x 1 x 2 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ = b 1 b 2 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ 1 3 0 1 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ x 1 x 2 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ = b 1 b 2 2 b 1 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ 10 6 3 1 7 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ x 1 x 2 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ = b 1 b 2 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ 10 6 3 0 7 3 x 10 6 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ x 1 x 2 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ = b 1 b 2 10 6 b 1 Ρਟ Σਿ ΢ਯ Τ੏ Φ੯ Υ੟ Generated a very large number Partial pivoting: use the largest number in the column (in absolute value)
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Nonsigularity and Singularity The (square) n x n matrix A is nonsingular if: Inverse A -1 exists det(A)
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