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Chem 302 - 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 ×, 6 + 10 th order 3628800 ×, 3628800 + 600 th order 1.27×10 1408 ×, 1.27×10 1408 +

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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 2 1297 3 482827 4 2406267 5 1440115133 10 798336008051063 20 1.0×10 20 59108323 100 1.9×10 160 6815501009603 1000 4.0×10 2570 6.7×10 8 1.0×10 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 10 0.16s1.6×10 -6 s2.1×10 -6 s 20 6475 years (2.4 days)1.2×10 -5 s1.7×10 -5 s 100 1×10 144 (1×10 138 ) years1.4×10 -3 s2.0×10 -3 s 1000 10 2554 (10 2548 ) 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 iterationx1x2x3 0000 110.5714291.333333 21.095238 1.047619 30.9940481.0272110.968254 40.992630.9900790.998299 51.0010270.9984611.00274 61.0005351.000930.999943 70.9998771.000060.999778 80.9999650.9999191.000021 91.0000131.0000011.000017 101.0000021.0000070.999997 110.999999 1210.9999991 13111

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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 iterationx1x2x3 0000 110.5714291.333333 21.095238 1.047619 30.9940481.0272110.968254 40.992630.9900790.998299 51.0010270.9984611.00274 61.0005351.000930.999943 70.9998771.000060.999778 80.9999650.9999191.000021 91.0000131.0000011.000017 101.0000021.0000070.999997 110.999999 1210.9999991 13111 Gauss-Seidel Method iterationx1x2x3 0000 110.7142861.031746 21.0396831.0147390.989544 30.9968510.9965631.001082 41.0005651.000390.999831 50.999930.9999421.000022 61.000011.0000080.999997 70.999999 1 8111 9111

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

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