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ECE 595, Section 10Numerical Simulations

Lecture 5: Linear Algebra

Prof. Peter Bermel

January 16, 2013

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Outline

Recap from Monday

Overview: Computational Linear Algebra

Gaussian Elimination LU Decomposition

Singular Value Decomposition

Sparse Matrices QR Decomposition

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Recap from Monday

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Overview: Computational Linear

Algebra

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Gauss-Jordan Elimination: No Pivoting

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Gauss-Jordan Elimination: Pivoting

Pivoting allows one more flexibility

Interchange rows (partial pivoting)

Interchange rows and columns (full pivoting)

Goal: put biggest element on diagonal

Caveat: some (artificial?) exceptions

Solution: implicit pivoting

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Gaussian Elimination

Reduce original matrix to partially empty (e.g.,

on lower left) to be called U

Perform backsubstitution to find the solution:

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LU Decomposition

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LU Decomposition

To construct the LU matrices, use Croutsalgorithm to compute the decomposition in

place:

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LU Decomposition

Execution time:

total number of elements computed: N2

Total number of operations per element (average):

N/3

Inversion: backsubstitute after factorization

Determinant of an LU factorization:

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Band Diagonal Matrices

Band diagonal means only k diagonals are

non-zero

Common special case: tridiagonal:

Can perform LU decomposition in kNoperations

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Iterative Improvement

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Singular Value Decomposition

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Sparse Linear Systems

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Band diagonal Block triangular Block tridiagonal

Other

Single-bordered

block diagonalDouble-bordered

block diagonal

Single-bordered

block triangular

Bordered band-

triangular

Single-bordered

band diagonal

Double-bordered

band diagonal

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Vandermonde Matrices

To solve the problem of moments, construct

Vandermonde matrices which look like:

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Cholesky Decomposition

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QR Decomposition

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Next Class

Discussion of root finding andoptimization

Please read Chapter 10 of NumericalRecipes by W.H. Press et al.

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