Chem 302 Math 252

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Chem 302 - Math 252

• Chapter 2Solutions 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

• Cramers Rule
• Gaussian Elimination
• Gauss-Jordan Elimination
• Maximum Pivot Strategy

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Cramers Rule

• Write coefficient matrix (A)
• Evaluate A
• If A0 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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Cramers Rule
Not singular System has unique solution
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Cramers Rule
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Cramers 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 (n1)! (n1)!
  Additions
• 2nd order 6 , 6
• 10th order 3628800 , 3628800
• 600th order 1.27101408 , 1.27101408

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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 ?n3n2- ?n , ?n3½n2- ?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 ½n3n2- 2½n2 , ½n3-1½n1
• 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 (nlt10) not a big deal
• Large systems critical

Number of floating point operations
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Comparison of Direct Methods
Time required on a 300 MFLOP computer (500 TFLOP)
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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
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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
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Indirect Methods

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