Chem 302 - Math 252 Chapter 2 Solutions of Systems of Linear Equations / Matrix Inversion.
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Transcript of Chem 302 - Math 252 Chapter 2 Solutions of Systems of Linear Equations / Matrix Inversion.
Chem 302 - Math 252
Chapter 2Solutions 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
Linear Equations 1 (Unique Solution)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.5 1 1.5 2 2.5 3 3.5
x
x-y=-1
2x+y=4
x=1, y=2
Linear Equations 2 (No Solution)
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5 3 3.5
x
2x+3y=6
4x+6y=10
Linear Equations 3 (Infinite Solutions)
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5 3 3.5
x
2x+3y=6
4x+6y=12
Linear Equations 4 (Almost Singular)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 0.5 1 1.5 2 2.5
x
7x+10y=175x+7y=12
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
1. Write coefficient matrix (A)
2. Evaluate |A|– If |A|=0 then singular
3. Form A1
– Replace column 1 of A with answer column
4. Compute x1 = |A1|/|A|
5. Repeat 3 and 4 for other variables
Cramer’s Rule1 2 3
1 2 3
1 2 3
2 7 4 9
9 6 1
3 8 5 6
x x x
x x x
x x x
2 7 4
1 9 6
3 8 5
A
2 7 4
1 9 6 235
3 8 5
A Not singular: System has unique
solution
1
7 4
9 6
8 5
9
1
6
A 1
9 7 4
1 9 6 940
6 8 5
A 1 1 / 940 / 235 4x A A
Cramer’s Rule
2
9
1
6
2 4
1 6
3 5
A 2 2 / 235/ 235 1x A A
3 3 / 470 / 235 2x A A
2
2 9 4
1 1 6 235
3 6 5
A
3
2 7
1 9
3 8
9
1
6
A 3
2 7 9
1 9 1 470
3 8 6
A
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
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 n3+n2- n ×, n3+½n2- n +
Gaussian Elimination1 2 3
1 2 3
1 2 3
2 7 4 9
9 6 1
3 8 5 6
x x x
x x x
x x x
2 7 4 9
| 1 9 6 1
3 8 5 6
A b
2 7 4 9
0 12.5 8 3.5
0 2.5 11 19.5
2 7 4 9
0 12.5 8 3.5
0 0 9.4 18.8
3
2 3
1 3 2
18.8/9.4 2
( 3.5 8 ) /12.5 1
(9 4 7 ) / 2 4
x
x x
x x x
Gauss-Jordan Elimination
1. Form augmented matrix2. Normalize 1st row3. 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 Elimination1 2 3
1 2 3
1 2 3
2 7 4 9
9 6 1
3 8 5 6
x x x
x x x
x x x
2 7 4 9
| 1 9 6 1
3 8 5 6
A b
7 92 21 2
1 9 6 1
3 8 5 6
1
2
3
4
1
2
x
x
x
7 92 2
25 72 2
5 392 2
1 2
0 8
0 11
7 92 2
16 725 25
5 392 2
1 2
0 1
0 11
6 8825 25
16 725 25
47 945 5
1 0
0 1
0 0
6 8825 25
16 725 25
1 0
0 1
0 0 1 2
1 0 0 4
0 1 0 1
0 0 1 2
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 Elimination1 2 3
1 2 3
1 2 3
9 6 1
2 7 4 9
3 8 5 6
x x x
x x x
x x x
1 9 6 1
| 2 7 4 9
3 8 5 6
A b
1 2 19 3 91
2 7 4 9
3 8 5 6
2
1
3
1
4
2
x
x
x
1 2 19 3 9
25 8829 3 9
35 31 469 3 9
1
0
0
1 2 19 3 9
25 8829 3 9
35 4693 93
1
0
0 1
13 4193 93
235 94093 93
35 4693 93
1 0
0 0
0 1
13 4193 93
35 4693 93
1 0
1 0 0 4
0 1
0 1 0 1
1 0 0 4
0 0 1 2
Comparison of Direct Methods
• Small systems (n<10) not a big deal
• Large systems criticalNumber 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 MethodsTime 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 2
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
11 1 12 2 1 1
21 1 22 2 2 2
1 1 2 2
n n
n n
n n nn n n
a x a x a x b
a x a x a x b
a x a x a x b
1 1 12 2 13 3 111
2 2 21 1 23 3 122
1 1 2 2 1 1
1
1
1
n n
n n
n n n n nn nnn
x b a x a x a xa
x b a x a x a xa
x b a x a x a xa
Jacobi Method
1 2 3
1 2 3
1 2 3
8 8
7 2 4
2 1 9 12
x x x
x x x
x x x
1 11 2 38 8
12 1 37
13 1 29
1
4 2
12 2
x x x
x x x
x x x
iteration x1 x2 x3
0 0 0 0
1 1 0.571429 1.333333
2 1.095238 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 0.999999 0.999999
12 1 0.999999 1
13 1 1 1
Gauss-Seidel Method
• Same as Jacobi method, but use updated values as soon as they are calculated.
Jacobi Method
1 11 2 38 8
12 1 37
13 1 29
1
4 2
12 2
x x x
x x x
x x x
iteration x1 x2 x3
0 0 0 0
1 1 0.571429 1.333333
2 1.095238 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 0.999999 0.999999
12 1 0.999999 1
13 1 1 1
Gauss-Seidel Method
iteration x1 x2 x3
0 0 0 0
1 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 0.999999 1
8 1 1 1
9 1 1 1
Indirect Methods
• Sufficient condition– Diagonally dominant
• Large problems
• Sparse matrix (many zeros)