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Chem Math 252 Chapter 2 Solutions of Systems of Linear Equations / Matrix Inversion

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Solutions of Systems of Linear Equations n linear equations, n unknowns Three possibilities –Unique solution –No solution –Infinite solutions Numerically systems that are almost singular cause problems –Range of solutions –Ill-conditioned problem Singular Systems

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Solutions of Systems of Linear Equations Direct Methods –Determine solution in finite number of steps –Usually preferred –Round-off error can cause problems Indirect Methods –Use iteration scheme –Require infinite operations to determine exact solution –Useful when Direct Methods fail

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Direct Methods Cramer’s Rule Gaussian Elimination Gauss-Jordan Elimination –Maximum Pivot Strategy

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Cramer’s Rule 1.Write coefficient matrix (A) 2.Evaluate |A| –If |A|=0 then singular 3.Form A 1 –Replace column 1 of A with answer column 4.Compute x 1 = |A 1 |/|A| 5.Repeat 3 and 4 for other variables

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Cramer’s Rule Not singular: System has unique solution

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Cramer’s Rule

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Good for small systems Good if only one or two variables are needed Very slow and inefficient for large systems –n order system requires (n+1)! × & (n+1)! Additions 2 nd order 6 ×, th order ×, th order 1.27× ×, 1.27×

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Gaussian Elimination 1.Form augmented matrix 2.Use elementary row operations to transform the augmented matrix so that the A portion is in upper triangular form Switch rows Multiply row by constant Linear combination of rows 3.Use back substitution to find solutions Requires n 3 +n 2 - n ×, n 3 +½n 2 - n +

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Gaussian Elimination

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Gauss-Jordan Elimination 1.Form augmented matrix 2.Normalize 1 st row 3.Use elementary row operations to transform the augmented matrix so that the A portion is the identity matrix Switch rows Multiply row by constant Linear combination of rows Requires ½n 3 +n 2 - 2½n+2 ×, ½n 3 -1½n+1 + Can also be used to find matrix inverse

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Gauss-Jordan Elimination

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Maximum Pivot Strategy Elimination methods can run into difficulties if one or more of diagonal elements is close to (or exactly) zero Normalize row with largest (magnitude) element.

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Gauss-Jordan Elimination

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Comparison of Direct Methods Small systems (n<10) not a big deal Large systems critical Number of floating point operations nCramer’sGaussian EliminationGauss-Jordan Elimination × × × × ×10 9

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Comparison of Direct Methods Time required on a 300 MFLOP computer (500 TFLOP) nCramer’sGaussian EliminationGauss-Jordan Elimination 2 2.4×10 -8 s1.8×10 -8 s1.4×10 -8 s 3 9.6×10 -8 s5.6×10 -8 s5.4×10 -8 s 4 4.8×10 -7 s1.2×10 -7 s1.3×10 -7 s 5 2.9×10 -6 s2.3×10 -7 s2.7×10 -7 s s1.6×10 -6 s2.1×10 -6 s years (2.4 days)1.2×10 -5 s1.7×10 -5 s 100 1× (1× ) years1.4×10 -3 s2.0×10 -3 s ( ) years1.32

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Indirect Methods Jacobi Method Gauss-Seidel Method Use iterations –Guess solution –Iterate to self consistent Can be combined with Direct Methods

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Jacobi Method Rearrange system of equations to isolate the diagonal elements Guess solution Iterate until self-consistent

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Jacobi Method iterationx1x2x

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Gauss-Seidel Method Same as Jacobi method, but use updated values as soon as they are calculated.

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Jacobi Method iterationx1x2x Gauss-Seidel Method iterationx1x2x

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Indirect Methods Sufficient condition –Diagonally dominant Large problems Sparse matrix (many zeros)

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