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Computational Physics Linear Algebra

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Computational PhysicsLinear Algebra

Dr. Guy Tel-Zur

Sunset in Caruaru by Jaime JaimeJunior. publicdomainpictures.net

Version 4-11-10, 14:00

- In this talk we deal with basic matrix operations
- Such as the solution of linear equations, calculate the
inverse of a matrix, its determinant etc.

- Here we focus in particular on so-called director 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.

- 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

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.

EISPACK 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.

LINPACK 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

- 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.

http://icl.cs.utk.edu/lapack-for-windows/

Demo

Work dir: C:\Users\telzur\Documents\BGU\Teaching\ComputationalPhysics\2011A\Lectures\04\LAPACK\CLAPACK-EXAMPLE\Release>

Demo

clear all

close all

disp('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

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’

http://ocw.mit.edu/courses/mathematics/18-085-computational-science-and-engineering-i-fall-2008/

What are the properties of K ?

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!!!!

- These matrices are symmetric.
- The matrices Kn are sparse.
- These matrices are tridiagonal.
- The matrices have constant diagonals.
- All the matrices K = Kn are invertible.
- 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

- (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.

Source: Computational Science and Engineering, by Gilbert Strang

T=Top, B=Bottom

Source: Computational Science and Engineering, by Gilbert Strang

Demo

Make a demo also using Octave

Source: Computational Science and Engineering, by Gilbert Strang

Demo

K4=toeplitz([2 -1 0 0])

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

inv(K4) …OK

inv(C4) …not OK singular

eig(K4) positive >0 positive definite

eig(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) …OK

inv(B4) …not OK singular

Demo

e=ones(8,1)

B.e=0 . = dot product

BT=transpose(B) B is symmetric

System to solve: Bu=f

uTBT=fT

BT.e=B.e=0

Thefore

fT.e=0

Demo

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

Demo

THIS IS THE CODE TO CREATE K,T,B,C AS SPARSE MATRICES

Then 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 Kif nargin<1, type='K'; endif nargin<2, n=10; ende=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.');endif nargin<3 | ~sparse K=full(K);end

http://math.mit.edu/classes/18.085/create-sparse.html

Source: Numerical recipes in FORTRAN77: the art of scientific computing, page 65. By William H. Press

In Mathematics: Vectors and Matrices

Are mapped to Computers as Memory arrays

Fixed Memory allocation vs. Dynamic Memory Allocation

Compile time vs. Run time

In the next slides we will study two programs that demonstrate 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() {

intk,m, row = 3, col = 5;

intvec[5]; // line a: a standard C++ declaration of a vector

intmatr[3][5]; // line b: a standard fixed-size C++ declaration of a matrix

for(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 data

for(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]);

} // הקוד שכתוב בספר קצת רשלני. לא ניתן להעתיק אותו אחד לאחד

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, intvec[], intmatr[][5]) { //line d: a func. def.

intk,m;

printf("\n Vetor data in sub1():\n"); // print vector data

for(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()

Table 4.2

Demo

equiv to int *vec

equiv to

int (*matr)[5]

Make demo in class

התכנית ב 4.3 לא מתקמפלת ומלאה באגים.

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

#include <stdio.h>

int main()

{

int *vec; // line a

int **matr; // line b

int m, k, row, col, total = 0;

printf("\n Read in number of rows= "); // line c

scanf("%d",&row);

printf("\n Read in number of column= ");

scanf("%d", &col);

vec = new int [col]; // line d

matr = (int **)matrix(row, col, sizeof(int)); // line e

for(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 data

for(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 array

for(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 f

delete [] vec; // line g

return 0;

} // End: function main()

we declare a pointer to an integer

declares a pointer-to-a-pointer which will contain the address to

a 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

we use a user-defined function to reserve

necessary memory for matrix[row][col] and again matr contains the address to the reserved memory

location.

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

the same procedure is performed for vec[]

Continued from previous slide

void sub_1(int row, int col, intvec[], int ??matr) // line h

{

intk,m;

printf("\n Vetor data in sub1():\n"); // print vector data

for(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()

in line h an important difference from the previous case occurs. First, the vector declaration is

the 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

/*

* 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

* intnum_bytes - number of bytes for each element

* Returns avoid **pointer to the reserved memory location.

*/

void **matrix( int row , int col , intnum_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 ( )

lib.cpp

Skip to section 4.4

If we store a matrix in the sequence a11 a12 . . . a1n a21 a22 . . . a2n . . . ann

This 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,

- Gauss Elimination
- Upper/Lower triangular matrix
- Solve Linear System
- LU algorithm
- Cholesky decomposition (a special case)

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

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

Case #1:

Assume first that the function f does not depend on u(x).

Then our linear equation reduces to

Au = f , (4.8)

which is nothing but a simple linear equation with a tridiagonal matrix A.

We will solve such a system of equations in subsection 4.4.3.

Case #2

If we assume that our boundary value problem is that of a quantum mechanical particle confined by

a harmonic oscillator potential, then our function f takes the form (assuming that all constants m = h_bar = omega = 1

with λ being the eigenvalue.

Inserting this into our equation, we define first a new matrix A as

להבנה נוספת ראש השקף הבא

We will solve this type of equations in chapter 12.

LU Factorization

Continue from page 83 at MHJ chap 4 PDF

MHJ Chapter 4, page 87 (2009 edition)

One has to check first if A is positive definite!

MHJ Chap. 4, page 93

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

M=UΣV*

Matlab demo:

M=[1 0 0 0 2; 0 0 3 0 0; 0 0 0 0 0;0 4 0 0 0]

[U S V]=svd(M)