Chapter 2 Solutions of Systems of Linear Equations / Matrix Inversion Chem 302 - Math 252 Chapter 2 Solutions of Systems of Linear Equations / Matrix Inversion

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

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

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

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

Cramer’s Rule Not singular: System has unique solution

Cramer’s Rule

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 3628800 ×, 3628800 + 600th order 1.27×101408 ×, 1.27×101408 +

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 +

Gaussian Elimination

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

Gauss-Jordan Elimination

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.

Gauss-Jordan Elimination

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 79833600 805 1063 20 1.0×1020 5910 8323 100 1.9×10160 681550 1009603 1000 4.0×102570 6.7×108 1.0×109

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 102554 (102548) years 1.3

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

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

Jacobi Method iteration x1 x2 x3 1 0.571429 1.333333 2 1.095238 1 0.571429 1.333333 2 1.095238 1.047619 3 0.994048 1.027211 0.968254 4 0.99263 0.990079 0.998299 5 1.001027 0.998461 1.00274 6 1.000535 1.00093 0.999943 7 0.999877 1.00006 0.999778 8 0.999965 0.999919 1.000021 9 1.000013 1.000001 1.000017 10 1.000002 1.000007 0.999997 11 0.999999 12 13

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

Jacobi Method Gauss-Seidel Method iteration x1 x2 x3 1 0.571429 1 0.571429 1.333333 2 1.095238 1.047619 3 0.994048 1.027211 0.968254 4 0.99263 0.990079 0.998299 5 1.001027 0.998461 1.00274 6 1.000535 1.00093 0.999943 7 0.999877 1.00006 0.999778 8 0.999965 0.999919 1.000021 9 1.000013 1.000001 1.000017 10 1.000002 1.000007 0.999997 11 0.999999 12 13 iteration x1 x2 x3 1 0.714286 1.031746 2 1.039683 1.014739 0.989544 3 0.996851 0.996563 1.001082 4 1.000565 1.00039 0.999831 5 0.99993 0.999942 1.000022 6 1.00001 1.000008 0.999997 7 0.999999 8 9

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