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**Chapter 2 Solutions of Systems of Linear Equations / Matrix Inversion**

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 Write coefficient matrix (A) Evaluate |A| Form A1**

If |A|=0 then singular Form A1 Replace column 1 of A with answer column Compute x1 = |A1|/|A| 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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**Cramer’s Rule 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 2nd order 6 ×, 6 + 10th order ×, 600th order 1.27× ×, 1.27×

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**Gaussian Elimination Form augmented matrix**

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 Use back substitution to find solutions Requires n3+n2- n ×, n3+½n2- n +

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Gaussian Elimination

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**Gauss-Jordan Elimination**

Form augmented matrix Normalize 1st row 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 ½n3+n2- 2½n+2 ×, ½n3-1½n+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 n Cramer’s Gaussian Elimination Gauss-Jordan Elimination 2 12 9 7 3 48 28 27 4 240 62 67 5 1440 115 133 10 805 1063 20 1.0×1020 5910 8323 100 1.9×10160 681550 1000 4.0×102570 6.7×108 1.0×109

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**Comparison of Direct Methods**

Time required on a 300 MFLOP computer (500 TFLOP) n Cramer’s Gaussian Elimination Gauss-Jordan Elimination 2 2.4×10-8s 1.8×10-8s 1.4×10-8s 3 9.6×10-8s 5.6×10-8s 5.4×10-8s 4 4.8×10-7s 1.2×10-7s 1.3×10-7s 5 2.9×10-6s 2.3×10-7s 2.7×10-7s 10 0.16s 1.6×10-6s 2.1×10-6s 20 6475 years (2.4 days) 1.2×10-5s 1.7×10-5s 100 1×10144 (1×10138) years 1.4×10-3s 2.0×10-3s 1000 (102548) years 1.3

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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 iteration x1 x2 x3 1 0.571429 1.333333 2 1.095238**

1 2 3 4 5 6 7 8 9 10 11 12 13

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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 Gauss-Seidel Method iteration x1 x2 x3 1 0.571429**

1 2 3 4 5 6 7 8 9 10 11 12 13 iteration x1 x2 x3 1 2 3 4 5 6 7 8 9

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**Indirect Methods Sufficient condition Large problems**

Diagonally dominant Large problems Sparse matrix (many zeros)

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