Scientific ComputingLinear Algebra

Dr. Guy Tel-Zur

Sunset in Caruaru by Jaime JaimeJunior. publicdomainpictures.net Version 10-04-11, 14:00

MHJ Chapter 4 – Linear Algebra

• In this talk we deal with basic matrix operations• Such as the solution of linear equations, calculate theinverse of a matrix, its determinant etc. • Here we focus in particular on so-called direct or elimination

methods, which are in principle determined through a finite number of arithmetic operations.

• Iterative methods will be discussed in connection with eigenvalue problems in MHJ chapter 12.

• This chapter serves also the purpose of introducing important programming details such as handling memory allocation for matrices and the usage of the libraries which follow these lectures.

Libraries• LAPACK based on:

– EISPACK – for solving symmetric, un-symmetric and generalized eigenvalue problems

– LINPACK - linear equations and least square problems

• BLAS: Basic Linear Algebra Subprogram– Levels I, II and III

• Nelib: http://www.netlib.org

solu

tions

Build

ing

bloc

ks

LAPACK - Linear Algebra PACKage

LAPACK is written in Fortran90 and provides routines for solving systems of simultaneous linear equations, least-squares solutions of linear systems of equations, eigenvalue problems, and singular value problems. The associated matrix factorizations (LU, Cholesky, QR, SVD…) are also provided. Dense and banded matrices are handled, but not general sparse matrices. In all areas, similar functionality is provided for real and complex matrices, in both single and double precision.

EISPACKEISPACK is a collection of Fortran subroutines that compute the eigenvalues and eigenvectors of nine classes of matrices: complex general, complex Hermitian, real general, real symmetric, real symmetric banded, real symmetric tridiagonal, special real tridiagonal, generalized real, and generalized real symmetric matices. In addition, two routines are included that use singular value decomposition to solve certain least-squares problems.

LINPACKLINPACK is a collection of Fortran subroutines that analyze and solve linear equations and linear least-squares problems. The package solves linear systems whose matrices are general, banded, symmetric indefinite, symmetric positive definite, triangular, and tridiagonal square. In addition, the package computes the QR and singular value decompositions of rectangular matrices and applies them to least-squares problems. LINPACK uses column-oriented algorithms to increase efficiency by preserving locality of reference.

• LINPACK and EISPACK are based on BLAS I• LAPACK is based on BLAS III

BLAS• The BLAS (Basic Linear Algebra Subprograms) are routines

that provide standard building blocks for performing basic vector and matrix operations.

• The Level 1 BLAS perform scalar, vector and vector-vector operations.

• The Level 2 BLAS perform matrix-vector operations • The Level 3 BLAS perform matrix-matrix operations.• Because the BLAS are efficient, portable, and widely

available, they are commonly used in the development of high quality linear algebra software, LAPACK for example.

LAPACK / CLAPACK / ScaLAPACK for Windowshtt

p://

icl.c

s.ut

k.ed

u/la

pack

-for-w

indo

ws/

For all matrix/problem structures:Ex: LAPACK Table of Contents

• BD – bidiagonal• GB – general banded• GE – general • GG – general , pair• GT – tridiagonal• HB – Hermitian banded• HE – Hermitian• HG – upper Hessenberg, pair• HP – Hermitian, packed• HS – upper Hessenberg• OR – (real) orthogonal• OP – (real) orthogonal, packed• PB – positive definite, banded• PO – positive definite• PP – positive definite, packed• PT – positive definite, tridiagonal

• SB – symmetric, banded• SP – symmetric, packed• ST – symmetric, tridiagonal • SY – symmetric • TB – triangular, banded• TG – triangular, pair• TB – triangular, banded• TP – triangular, packed• TR – triangular• TZ – trapezoidal• UN – unitary• UP – unitary packed

Source: CS267 Lecture 13

An Example - DGESV• Ax=b• DGESV computes the solution to a real system of linear

equations A * X = B, where A is an N-by-N matrix and X and B are N-by-NRHS matrices.

• The LU decomposition with partial pivoting and row interchanges is used to factor A asA = P * L * U,where P is a permutation matrix, L is unit lower triangular, and U is upper triangular. The factored form of A is then used to solve the system of equations A * X = B.

ArgumentsN (input) INTEGER , The number of linear equations, i.e., the order of the matrix A. N >= 0. NRHS (input) INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. A (input/output) DOUBLE PRECISION array, dimension (LDA,N) On entry, the N-by-N coefficient matrix A. On exit, the factors L and U from the factorization A = P*L*U; the unit diagonal elements of L are not stored. LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,N). IPIV (output) INTEGER array, dimension (N) The pivot indices that define the permutation matrix P; row i of the matrix was interchanged with row IPIV(i). B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) On entry, the N-by-NRHS matrix of right hand side matrix B. On exit, if INFO = 0, the N-by-NRHS solution matrix X. LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). INFO (output) INTEGER = 0: successful exit< 0: if INFO = -i, the i-th argument had an illegal value> 0: if INFO = i, U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution could not be computed.

Working with a Linear Algebra package using Visual Studio

Demo Work dir: C:\Users\telzur\Documents\Weizmann\ScientificComputing\SC2011B\Lectures\04\LAPACK\CLAPACK-EXAMPLE\Release

Compare with the following MATLAB code

clear allclose alldisp('solves Ax=b')A= [76, 27, 18; 25, 89, 60; 11, 51, 32]b=[10, 7, 43] invA=inv(A) x=b*inv(A) [L,U]=lu(A) inv(U)*inv(L)*b'

Code location: C:\Users\telzur\Documents\BGU\Teaching\ComputationalPhysics\2011A\Lectures\04\lpack.m

Demo

DGESV SUBROUTINE DGESV( N, NRHS, A, LDA, IPIV, B, LDB, INFO )** -- LAPACK driver routine (version 3.2) --* -- LAPACK is a software package provided by Univ. of Tennessee, --* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--* November 2006** .. Scalar Arguments .. INTEGER INFO, LDA, LDB, N, NRHS* ..* .. Array Arguments .. INTEGER IPIV( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * )* ..** Purpose* =======** DGESV computes the solution to a real system of linear equations* A * X = B,* where A is an N-by-N matrix and X and B are N-by-NRHS matrices.** The LU decomposition with partial pivoting and row interchanges is* used to factor A as* A = P * L * U,* where P is a permutation matrix, L is unit lower triangular, and U is* upper triangular. The factored form of A is then used to solve the* system of equations A * X = B.*

* Arguments* =========** N (input) INTEGER* The number of linear equations, i.e., the order of the* matrix A. N >= 0.** NRHS (input) INTEGER* The number of right hand sides, i.e., the number of columns* of the matrix B. NRHS >= 0.** A (input/output) DOUBLE PRECISION array, dimension (LDA,N)* On entry, the N-by-N coefficient matrix A.* On exit, the factors L and U from the factorization* A = P*L*U; the unit diagonal elements of L are not stored.** LDA (input) INTEGER* The leading dimension of the array A. LDA >= max(1,N).** IPIV (output) INTEGER array, dimension (N)* The pivot indices that define the permutation matrix P;* row i of the matrix was interchanged with row IPIV(i).** B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)* On entry, the N-by-NRHS matrix of right hand side matrix B.* On exit, if INFO = 0, the N-by-NRHS solution matrix X.** LDB (input) INTEGER* The leading dimension of the array B. LDB >= max(1,N).** INFO (output) INTEGER* = 0: successful exit* < 0: if INFO = -i, the i-th argument had an illegal value* > 0: if INFO = i, U(i,i) is exactly zero. The factorization* has been completed, but the factor U is exactly* singular, so the solution could not be computed.** =====================================================================*

Cont’

* .. External Subroutines .. EXTERNAL DGETRF, DGETRS, XERBLA* ..* .. Intrinsic Functions .. INTRINSIC MAX* ..* .. Executable Statements ..** Test the input parameters.* INFO = 0 IF( N.LT.0 ) THEN INFO = -1 ELSE IF( NRHS.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -7 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DGESV ', -INFO ) RETURN END IF** Compute the LU factorization of A.* CALL DGETRF( N, N, A, LDA, IPIV, INFO ) IF( INFO.EQ.0 ) THEN** Solve the system A*X = B, overwriting B with X.* CALL DGETRS( 'No transpose', N, NRHS, A, LDA, IPIV, B, LDB, $ INFO ) END IF RETURN** End of DGESV* END

Cont’

LAPACK Benchmarkhtt

p://

ww

w.a

ts.u

cla.

edu/

clus

ters

/com

mon

/soft

war

e/lib

rarie

s/la

pack

_ben

chm

ark.

htm

MFLOPS is calculated as: MFLOPS = (1x10**-6) * [(2/3)*N**3 + 2*N**2] / (CPU seconds)

Cost

Solving A*x=b using GE

Factorize A = L*U using GE (cost = 2/3 n3 flops)

Solve L*y = b for y, using substitution (cost = n2 flops)

Solve U*x = y for x, using substitution (cost = n2 flops)PDGESV = ScaLAPACK Parallel LU

(“Linpack Benchmark”),

Mathematical intermezzo

http:

//oc

w.m

it.ed

u/co

urse

s/m

athe

mati

cs/1

8-08

5-co

mpu

tatio

nal-

scie

nce-

and-

engi

neer

ing-

i-fal

l-200

8/

The Kn Matrices

The next slides are from: Computational Science and Engineering, by Gilbert Strang. An excellent recommended book. His course is available online from MIT Opencourseware. Very Recommended!!!!

What are the properties of K ?

Kn Properties1. These matrices are symmetric.2. The matrices Kn are sparse.3. These matrices are tridiagonal.4. The matrices have constant diagonals.5. All the matrices K = Kn are invertible.6. The symmetric matrices Kn are positive

definite.

Source: Computational Science and Engineering, by Gilbert Strang

• In Signal Processing D=Kn/4 is a “High-Pass” Filter. Du picks out the rapidly varying parts of a vector u

• K are called Toeplitz Matrix and MATLAB has a function for generating such matrices, e.g. K = toeplitz([2 -1 zeros(l,2)]) constructs K4 from row 1

Source: Computational Science and Engineering, by Gilbert Strang

More properties• (Pivots) An invertible matrix has n nonzero

pivots.• A positive definite symmetric matrix has n

positive pivots.• (Eigenvalues) An invertible matrix has n nonzero

eigenvalues.• A positive definite symmetric matrix has n

positive eigenvalues. (Positive definite means a quadratic form that satisfies: XTAX>0 )

Source: Computational Science and Engineering, by Gilbert Strang

More special matrices

T=Top, B=Bottom

Source: Computational Science and Engineering, by Gilbert Strang

Building K,T,B,C in Matlab

Source: Computational Science and Engineering, by Gilbert Strang

Make a demo also using Octave

Demo

Matlab DemosK4=toeplitz([2 -1 0 0])

C4=toeplitz([2 -1 0 -1])

inv(K4) …OKinv(C4) …not OK singular

eig(K4) positive >0 positive definiteeig(C4) >=0 semi positive definite

Both are symmetric: try transpose(K4)

Check:[L,U]=lu(T4) all pivots are 1[L,U]=lu(B4) 0 on the diagonal of U B isn’t invertibe

inv(T4) …OKinv(B4) …not OK singular

Demo

demos

e=ones(8,1)

B.e=0 . = dot productBT=transpose(B) B is symmetric

System to solve: Bu=f

uTBT=fT

BT.e=B.e=0

TheforefT.e=0

Demo

Continued det(K8) = Π(diagonal of U) = 2/1 * 3/2 * 4/3 … * 9/8 = 9

Demo

For large size nTHIS IS THE CODE TO CREATE K,T,B,C AS SPARSE MATRICESThen Matlab will not operate on all the zeros! function K=SKTBC(type,n,sparse)% SKTBC Create finite difference model matrix.% K=SKTBC(TYPE,N,SPARSE) creates model matrix TYPE of size N-by-N.% TYPE is one of the characters 'K', 'T', 'B', or 'C'.% The command K = SKTBC('K', 100, 1) gives a sparse representation% K=SKTBC uses the defaults TYPE='K', N=10, and SPARSE=false.% Change the 3rd argument from 1 to 0 for dense representation!% If no 3rd argument is given, the default is dense% If no argument at all, KTBC will give 10 by 10 matrix K

if nargin<1, type='K'; endif nargin<2, n=10; end

e=ones(n,1);K=spdiags([-e,2*e,-e],-1:1,n,n);switch type  case 'K'  case 'T'   K(1,1)=1;  case 'B'   K(1,1)=1;   K(n,n)=1;  case 'C'   K(1,n)=-1;   K(n,1)=-1;  otherwise    error('Unknown matrix type.');end

if nargin<3 | ~sparse  K=full(K);end

http:

//m

ath.

mit.

edu/

clas

ses/

18.0

85/c

reat

e-sp

arse

.htm

l

Demo

Sour

ce: N

umer

ical

reci

pes i

n FO

RTRA

N77

: the

art

of

scie

ntific

com

putin

g, p

age

65.

By W

illia

m H

. Pre

ss

In Mathematics: Vectors and MatricesAre mapped to Computers as Memory arrays

Fixed Memory allocation vs. Dynamic Memory AllocationCompile time vs. Run time

In the next slides we will study two programs that address these issues

Let’s start with Table 4.2 in the book: Matrix handling program where arrays are defined at compilation time – Next slide

Use Visual Studio for the demo solution file under: chap4_static

int main() {int k,m, row = 3, col = 5;int vec[5]; // line a: a standard C++ declaration of a vectorint matr[3][5]; // line b: a standard fixed-size C++ declaration of a matrixfor(k = 0; k < col; k++) vec[k] = k; // data into vector[] for(m = 0; < row; m++) { // data into matr[][] for(k = 0; k < col ; k++) matr[m][k] = m + 10 * k; }printf("\n Vector data in main():\n”); // print vector datafor(k = 0; k < col; k++) printf("vetor[%d]= %d ",k, vec[k]);printf("\n Matrix data in main():");for(m = 0; m < row; m++) { printf(“\n”); for(k = 0; k < col; k++) printf("m atr[%d][%d]= %d ",m,k,matr[m][k]);}printf(“\n”);sub_1(row, col, vec, matr); // line c: transfer vec[] and matr[][] addresses to func. sub_1().return 0;} // End: function main()void sub_1(int row, int col, int vec[], int matr[][5]) { //line d: a func. def.int k,m;printf("\n Vetor data in sub1():\n"); // print vector datafor(k = 0; k < col; k++) printf("vetor[%d]= %d ",k, vec[k]);printf("\n Matrix data in sub1():");for(m = 0; m < row; m++) { printf(“\n”); for(k = 0; k < col; k++) { printf("matr[%d][%d]= %d ",m, k, matr[m][k]); }}printf(“\n”); } // End: function sub_1()

equiv to int *vec

equiv to int (*matr)[5]

Table 4.2

Demo

Using Visual Studio 2010

Make demo in class

Static program execution

Table 4.3: Matrix handling program with dynamic array allocation.

ב ומלאה 4.3התכנית מתקמפלת לאבאגים.

. בשלב לדוגמה כתכנית תתווסף היא בעתיד - מבלי קוד פסאודו כאל אליה נתייחס הזה

להריצה

we declare a pointer to an

integer

declares a pointer-to-a-pointer which will contain the address toa pointer of row vectors, each with col integers

read in the size of vec[] and matr[][] through the numbers

row and col

we reserve memory for the vector

#include <stdio.h>int main(){int *vec; // line aint **matr; // line bint m, k, row, col, total = 0;printf("\n Read in number of rows= "); // line cscanf("%d",&row);printf("\n Read in number of column= ");scanf("%d", &col);vec = new int [col]; // line dmatr = (int **)matrix(row, col, sizeof(int)); // line efor(k = 0; k < col; k++) vec[k] = k; // store data in vector[]for(m = 0; < row; m++) { // store data in array[][] for(k = 0; k < col; k++) matr[m][k] = m + 10 * k;}printf("\n Vetor data in main():\n"); // print vector datafor(k = 0; k < col; k++) printf("vetor[%d]= %d ",k,vec[k]);printf("\n Array data in main():");for(m = 0; m < row; m++) { printf("\n"); for(k = 0; k < col; k++) { printf("m atrix[%d][%d]= %d ",m, k, matr[m][k]); }}printf("\n");for(m = 0; m < row; m++) { // access the arrayfor(k = 0; k < col; k++) total += matr[m][k];}printf("\n Total= %d\n",total);sub_1(row, col, vec, matr);free_matrix((void **)matr); // line fdelete [] vec; // line greturn 0;} // End: function main()

we use a user-defined function to reservenecessary memory for matrix[row][col] and again matr contains the address to the reserved memorylocation.

The remaining part of the function main() are as in the previous case down to line f.

the same procedure is performed for vec[]

void sub_1(int row, int col, int vec[], int ??matr) // line h{int k,m;printf("\n Vetor data in sub1():\n"); // print vector datafor(k = 0; k < col; k++) printf("vetor[%d]= %d ",k, vec[k]);printf("\n Matrix data in sub1():");for(m = 0; m < row; m++) { printf("\n"); for(k = 0; k < col; k++) { printf("matrix[%d][%d]= %d ",m,k,matr[m][k]); }}printf("\n");} // End: function sub_1()

Continued from previous slide

in line h an important difference from the previous case occurs. First, the vector declaration isthe same, but the matr declaration is quite different. The corresponding parameter in the call to sub_1[]in line g is a double pointer. Consequently, matr in line h must be a double pointer

lib.cpp

/** The function* void **matrix( )* reserves dynamic memory for a two-dimensional matrix* using the C++ command new . No initialization of the elements .* Input data :* int row - number of rows* int col - number of columns* int num_bytes - number of bytes for each element* Returns avoid **pointer to the reserved memory location.*/void **matrix( int row , int col , int num_bytes ){int i , num;char **pointer, *ptr;pointer = new(nothrow) char* [ row ] ;if ( !pointer ) {cout << "Exeption handling Memory aloation failed";cout << "for"<< row << "row addresses!"<< endl;return NULL;}i = ( row * col * num_bytes ) / size of ( char ) ;pointer [ 0 ] = new( nothrow ) char [ i ] ;if ( !pointer [ 0 ] ) {cout << "Exeption handling :Memory allocation failed";cout << "for address to " << i << " characters!"<< endl ;return NULL;}ptr = pointer [ 0 ] ;num = col * num_bytes ;for ( i = 0 ; i < row ; i ++ , ptr += num ) {pointer [ i ] = ptr ;}return ( void **) pointer ;} // end : function void **matrix ( )

Skip to section 4.4

C++ and Fortran features of matrix handling

If we store a matrix in the sequence a11 a12 . . . a1n a21 a22 . . . a2n . . . annThis is called “row-major” order (we go along a given row i and pick up all column elements j) C++ stores them by row-major.

We cloud also store in column-major order a11 a21 . . . an1 a12 a22 . . . an2 . . . ann.Fortran stores matrices by column-major,

4.4 Linear Systems• Gauss Elimination• Upper/Lower triangular matrix• Solve Linear System• LU algorithm• Cholesky decomposition (a special case)• QR

(4.4) Linear Systems: An Example

…And we switched from a Differential Equation to Linear Algebra.

This is the stiffness matrix, K, we met earlier!

LU Decomposition (Factorization)

Theorem: if the n-by-n matrix A is non singular, there exists a permutation matrix P (the identity matrix with its rows permuted), a non-singular lower triangular matrix L and a non-singular upper triangular matrix U such that: A=PLU. To solve Ax=b, we solve the equivalent PLUx=b as follows:

LUx=P-1b=PTb (permute entries of b)Ux=L-1 (PTb) (forward substitution)x=U-1 (L-1PTb) (backward substitution)

For further reading: Applied Numerical Linear Algebra by James Demmel (SIAM)

LU Decomposition (Factorization)

Reference: MHJ chap 4 (section 4.4.2)

det{L}=1

Cholesky’s FactorizationMHJ Chapter 4, page 87 (2009 edition)

One has to check first if A is positive definite!

Linear System of Equations

Ly

Ux

The Determinant of a Matrix

Cholesky Factorization

In MATLAB:

R=chol(K)

Then,R’R=K

[R,p]=chol(K)P = 0 for positive definite K

QR Factorization

In Matlab:

[q,r]=qr(M)

M is m-by-n

r is m-by-n upper triangular matrixAnd q is m-by-m unitary matrix

For real matrix q, q’q=I

And det{r}=det{M}

is a decomposition of the matrix into an orthogonal and an upper triangular matrix.

Vandermonde matrix

Singular Value Decomposition SVD

Syntaxs = svd(X)[U,S,V] = svd(X)

Description: The svd command computes the matrix singular value decomposition.

s = svd(X) returns a vector of singular values.

[U,S,V] = svd(X) produces a diagonal matrix S of the same dimension as X, with nonnegative diagonal elements in decreasing order, and unitary matrices U and V so that

X = U*S*V'.

VTV=I, UTU=ISi are called singular values, s1≥s2 ≥ … ≥ 0

Example

For the matrixX = 1 2 3 4 5 6 7 8

the statement[U,S,V] = svd(X)

producesU = -0.1525 -0.8226 -0.3945 -0.3800 -0.3499 -0.4214 0.2428 0.8007 -0.5474 -0.0201 0.6979 -0.4614 -0.7448 0.3812 -0.5462 0.0407S = 14.2691 0 0 0.6268 0 0 0 0V = -0.6414 0.7672 -0.7672 -0.6414