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Numerical Analysis

Table of Contents

0 Introduction

1 Error Analysis

2 Numeric Representation 3 Iteration

4 Linear Algebra

4.1 PLU Decomposition

4.2 PLU Decomposition on Tridiagonal Matrices

4.3 Positive-definite Matrices

4.4 Cholesky Decomposition

4.5 QR Decomposition

4.6 Eigenvalues and Eigenvectors

4.7 Norms

4.8 Ill-conditioned Matrices

4.9 The Jacobi Method

4.10 The Gauss-Seidel Method

4.11 Overrelaxation Techniques

5 Interpolation

6 Least Squares

7 Taylor Series

8 Bracketing

9 The Five Techniques

10 Root Finding

11 Optimization

12 Differentiation

13 Integration

14 Initial-value Problems

15 Boundary-value Problems

http://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/https://ece.uwaterloo.ca/Home/https://ece.uwaterloo.ca/Undergrad/https://ece.uwaterloo.ca/~dwharder/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/Contents/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/00Introduction/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/01Error/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/02Numerics/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/03Iteration/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/lup/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/tri/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/posdef/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/cholesky/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/qr/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/eigen/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/norms/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/illconditioned/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/jacobi/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/gauss/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/overrelax/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/05Interpolation/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/06LeastSquares/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/07TaylorSeries/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/08Bracketing/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/09TheFiveTechniques/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/10RootFinding/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/11Optimization/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/12Differentiation/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/13Integration/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/14IVPs/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/15BVPs/https://ece.uwaterloo.ca/Home/https://ece.uwaterloo.ca/Undergrad/https://ece.uwaterloo.ca/~dwharder/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/Contents/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/00Introduction/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/01Error/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/02Numerics/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/03Iteration/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/lup/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/tri/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/posdef/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/cholesky/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/qr/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/eigen/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/norms/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/illconditioned/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/jacobi/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/gauss/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/04LinearAlgebra/overrelax/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/05Interpolation/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/06LeastSquares/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/07TaylorSeries/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/08Bracketing/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/09TheFiveTechniques/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/10RootFinding/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/11Optimization/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/12Differentiation/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/13Integration/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/14IVPs/https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/15BVPs/http://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/

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Appendices

4.3 Positive-definite Matrices

Introduction Theory HOWTO Error Analysis Examples Questions Applications in Engineering Matlab Maple

Introduction

Positive-definite matrices have some nice properties which we will exploit in finding more convenient

decompositions which require less memory.

Background

Useful background for this topic includes:

Matrix multiplication and inner (dot) products

References

Bradie, Section 3.7, Special Matrices, p.211.

Weisstein,http://mathworld.wolfram.com/PositiveDefiniteMatrix.html.

Theory

A symmetric n n matrix M is said to be positive definite if for all nonzero vectors v, the product

vTMv > 0.

Looking more closely at this product, we see that it is the dot product ofvT and Mv. You will recall

that the dot product of two vectors is equal tovTMv = ||v||2 ||Mv||2 cos(),where is the angle

between v and Mv, and and thus, for the dot product to be positive, it means that the image ofv

must have an angle less than 90o, i.e., || < /2.

It is unreasonable to test every possible vector v to determine if a matrix is positive definite,

however, there are some cases which appear quite often in engineering where it is quite simple to

state that a matrix is positive definite. These are covered on the howtos. The following are some

interesting theorems related to positive definite matrices:

Theorem 4.2.1

A matrix is invertible if and only if all of the eigenvalues are non-zero.

Proof: Please refer to your linear algebra text.

Theorem 4.2.2A positive definite matrix M is invertible.

Proof: if it was not, then there must be a non-zero vectorx such that Mx = 0. ThereforexTMx = 0

which contradicts our assumption aboutM being positive definite.

Theorem 4.2.3

All the eigenvalues with corresponding real eigenvectors of a positive definite matrix M are positive.

https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/Appendices/http://mathworld.wolfram.com/PositiveDefiniteMatrix.htmlhttps://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/Appendices/http://mathworld.wolfram.com/PositiveDefiniteMatrix.html

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Proof: ifx is an eigenvector ofM then Mx = xand therefore xTMx = ||x||2. For this to be positive,

it follows that > 0.

HOWTO

Determining Positive-definiteness

A symmetric matrix is positive definite if:

1. all the diagonal entries are positive, and

2. each diagonal entry is greater than the sum of the absolute values of all other entries in the

corresponding row/column.

An arbitrary symmetric matrix is positive definite if and only ifeach of its principal submatrices has

a positive determinant. This is known as Sylvester's criterion.

Note that only the last case does the implication go both ways.

Diagonal Dominance

A matrix M is row diagonally dominantif

for i= 1, ..., n, column diagonally dominantif

for i= 1, ..., n, and diagonally dominant if it is both row and column diagonally dominant.

Thus, we may restate our first definition as:

A symmetric row diagonally-dominant matrix with positive diagonal entries is positive definite.

Error Analysis

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Examples

Example 1

Show that ifA is invertible, then ATA is positive definite.

IfA is invertible, then Av 0 for any vector v 0. Therefore vT(ATA)v = (vTAT)(Av) which is the

vectorAv dotted with itself, that is, the square of the norm (or length) of the vector. As Av 0, the

norm must be positive, and thereforevT(ATA)v > 0.

Questions

Question 1

Create a random positive-definite n n matrix in which is also diagonally dominant.

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( rand(n, n) + (n - 1)*eye( n ) )

Question 3

Which of the following matrices are positive definite?

a.

b.

c.

Applications to Engineering

The conductance matrix of a RLC circuit is positive definite. Additionally, we will see that the matrix

defined when performing least-squares fitting is also positive definite.

As an alternate example, the Hurwitz criteria for the stability of a differential equation requires that

the constructed matrix be positive definite. This is seen in signals and linear systems.

MatlabA test for positive definiteness requires that the matrix is symmetric and that all the eigenvalues are

positive.

all( all( M == M' ) ) & min( eig( M ) ) > 0

Alternatively, one may use the test

M = [...]; % assume M is square

isposdef = true;

for i=1:length(M)

if ( det( M(1:i, 1:i) )

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Douglas Wilhelm HarderDepartment of Electrical and Computer Engineering

University of Waterloo200 University Avenue WestWaterloo, Ontario, CanadaN2L 3G1Phone: +1 519 888 4567 extension 37023Fax: +1 519 746 3077

Contact the Author

Department of Electrical and Computer Engineering

Faculty of Engineering

mailto:dwharder@uwaterloo.cahttp://ece.uwaterloo.ca/http://engineering.uwaterloo.ca/http://www.corel.com/http://www.firefox.com/http://www.vim.org/https://ece.uwaterloo.ca/Computing/Logo/http://www.eng.uwaterloo.ca/~engdev/mailto:dwharder@uwaterloo.cahttp://ece.uwaterloo.ca/http://engineering.uwaterloo.ca/