Engineering Analysis ENG 3420 Fall 2009

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Engineering Analysis ENG 3420 Fall 2009. Dan C. Marinescu Office: HEC 439 B Office hours: Tu-Th 11:00-12:00. Lecture 17. Reading assignment Chapters 10 and 11, Linear Algebra ClassNotes Last time: Symmetric matrices; Hermitian matrices. Matrix multiplication Today:

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### Engineering Analysis ENG 3420 Fall 2009

Dan C. Marinescu

Office: HEC 439 B

Office hours: Tu-Th 11:00-12:00

Lecture 17

Reading assignment Chapters 10 and 11, Linear Algebra ClassNotes

Last time:

Symmetric matrices; Hermitian matrices.

Matrix multiplication

Today:

Linear algebra functions in Matlab

The inverse of a matrix

Vector products

Tensor algebra

Characteristic equation, eigenvectors, eigenvalues

Norm

Matrix condition number

Next Time

More on

LU Factorization

Cholesky decomposition

Lecture 17

2

Matrix analysis in MATLAB

Norm Matrix or vector norm

normest Estimate the matrix 2-norm

rank Matrix rank

det Determinant

trace Sum of diagonal elements

null Null space

orth Orthogonalization

rref Reduced row echelon form

subspaceAngle between two subspaces

Eigenvalues and singular values

eig Eigenvalues and eigenvectors

svd Singular value decomposition

eigs A few eigenvalues

svds A few singular values

poly Characteristic polynomial

polyeig Polynomial eigenvalue problem

condeig Condition number for eigenvalues

hess Hessenberg form

qz QZ factorization

schur Schur decomposition

Matrix functions

Expm Matrix exponential

Logm Matrix logarithm

Sqrtm Matrix square root

Funm Evaluate general matrix function

Linear systems of equations

\ and / Linear equation solution

inv Matrix inverse

cond Condition number for inversion

condest 1-norm condition number estimate

chol Cholesky factorization

cholinc Incomplete Cholesky factorization

linsolve Solve a system of linear equations

lu LU factorization

ilu Incomplete LU factorization

luinc Incomplete LU factorization

qr Orthogonal-triangular decomposition

lsqnonneg Nonnegative least-squares

pinv Pseudoinverse

lscovLeast squares with known covariance

The inverse of a square
• If [A] is a square matrix, there is another matrix [A]-1, called the inverse of [A], for which [A][A]-1=[A]-1[A]=[I]
• The inverse can be computed in a column by column fashion by generating solutions with unit vectors as the right-hand-side constants:
Canonical base of an n-dimensional vector space

100……000

010……000

001……000

…………….

000…….100

000…….010

000…….001

Matrix Inverse (cont)
• LU factorization can be used to efficiently evaluate a system for multiple right-hand-side vectors - thus, it is ideal for evaluating the multiple unit vectors needed to compute the inverse.
The response of a linear system
• The response of a linear system to some stimuli can be found using the matrix inverse.
Distance and norms
• Metric space   a set where the ”distance” between elements of the set is defined, e.g., the 3-dimensional Euclidean space. The Euclidean metric defines the distance between two points as the length of the straight line connecting them.
• A norm real-valued function that provides a measure of the size or “length” of an element of a vector space.
Vector Norms
• The p-norm of a vector X is:
• Important examples of vector p-norms include:
Matrix Norms
• Common matrix norms for a matrix [A] include:
• Note - max is the largest eigenvalue of [A]T[A].
Matrix Condition Number
• The matrix condition number Cond[A] is obtained by calculating Cond[A]=||A||·||A-1||
• In can be shown that:
• The relative error of the norm of the computed solution can be as large as the relative error of the norm of the coefficients of [A] multiplied by the condition number.
• If the coefficients of [A] are known to t digit precision, the solution [X] may be valid to onlyt-log10(Cond[A]) digits.
Built-in functions to compute norms and condition numbers
• norm(X,p)  Compute the p norm of vector X, where p can be any number, inf, or ‘fro’ (for the Euclidean norm)
• norm(A,p)  Compute a norm of matrix A, where p can be 1, 2, inf, or ‘fro’ (for the Frobenius norm)
• cond(X,p) or cond(A,p)  Calculate the condition number of vector X or matrix A using the norm specified by p.
LU Factorization
• LU factorization involves two steps:
• Decompose the [A] matrix into a product of:
• a lower triangular matrix [L] with 1 for each entry on the diagonal.
• and an upper triangular matrix [U
• Substitution to solve for {x}
• Gauss elimination can be implemented using LU factorization
• The forward-elimination step of Gauss elimination comprises the bulk of the computational effort.
• LU factorization methods separate the time-consuming elimination of the matrix [A] from the manipulations of the right-hand-side [b].
Gauss Elimination as LU Factorization
• To solve [A]{x}={b}, first decompose [A] to get [L][U]{x}={b}
• MATLAB’s lu function can be used to generate the [L] and [U] matrices: [L, U] = lu(A)
• Step 1  solve [L]{y}={b}; {y} can be found using forward substitution.
• Step 2 solve [U]{x}={y}, {x} can be found using backward substitution.
• In MATLAB: [L, U] = lu(A) d = L\b x = U\d
• LU factorization  requires the same number of floating point operations (flops) as for Gauss elimination.
• Advantage  once [A] is decomposed, the same [L] and [U] can be used for multiple {b} vectors.
Cholesky Factorization
• A symmetric matrix a square matrix, A, that is equal to its transpose:

A = AT (T stands for transpose).

• The Cholesky factorization based on the fact that a symmetric matrix can be decomposed as:

[A]= [U]T[U]

• The rest of the process is similar to LU decomposition and Gauss elimination, except only one matrix, [U], needs to be stored.
• Cholesky factorization with the built-in chol command:U = chol(A)
• MATLAB’s left division operator \ examines the system to see which method will most efficiently solve the problem. This includes trying banded solvers, back and forward substitutions, Cholesky factorization for symmetric systems. If these do not work and the system is square, Gauss elimination with partial pivoting is used.