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SS 4062/JS 3067 Numerical Methods IV 15 Lectures
Dr. Charles Patterson
[email protected] 2.48 Lloyd Building
I
Gauss elimination and LU decomposition
Introduction to the Gnu Scientific Library (GSL)
GSL handling of vectors and matrices
LU decomposition using GSL
Introduction to LAPACK
LU decomposition using DGETRF
II
Vector spaces and linear equations
Systems of m equations in n unknowns
Rank of a matrix
III
Linear dependence, bases and dimension
IV
The four fundamental spaces of linear algebra
Column space, null space, row space and left null space
The fundamental theorem of linear algebra
Existence of inverses
V
Orthogonality
Orthogonal subspaces
Second fundamental theorem of linear algebra
Matrices and subspaces
VI
Inner products and projections onto lines
Transpose of a matrix
Orthogonal bases, matrices and Gram-Schmidt orthogonalisation
VII
QR decomposition
Householder transformation
QR factorisation by Householder transformation
VIII
Eigenvalues and eigenvectors
Reduction to Hessenberg form and QR factorisation
Inverse power method for inverses
Singular value decomposition
Norm and condition number of a matrix
IX
Special functions
 and incomplete  functions
Bessel functions
Legendre polynomials
Course texts:
Linear algebra and its applications, 3rd Edn. Strang (Harcourt, Brace, Jovanovich)
LAPACK User’s Guide 3rd Edn. (SIAM)
Computational Physics Landau and Paez (Wiley)
I Gauss Elimination and LU Decomposition
Consider the system of linear equations
2u + v + w = 5
4u – 6v
= -2
-2u + 7v +2w = 9
(1)
(2)
(3)
Which can also be written in matrix form A.x = b
1 1  u   5 
 2

   
 4  6 0  v     2 
  2 7 2  w   9 

   
If we take a geometrical point of view, each of these equations corresponds to a plane
in the 3-D {u,v,w} Cartesian space. It is relatively easy to see that a unique solution
to the problem exists if all three planes meet at a single point – which corresponds to
the solution to the simultaneous equations. Alternatively, two planes may be parallel
and not intersect or the 3 planes may meet in pairs along lines, or all 3 planes may
meet along a line.
If we rewrite the set of equations in terms of the columns of the matrix A then we can
gain some insight into what kind of solution is possible, if any.
 2   1 
1  5 
   
   
u 4   v  6   w 0     2 
  2  7 
 2  9 
   
   
We see that the vector on the rhs must be constructed from a linear combination of 3
vectors on the lhs whose coefficients are u, v and w.
Consider the example
u + v + w =2
2u
+ 3w = 5
3u + v + 4w = 6
(4)
(5)
(6)
In this case the lhs of (4)+(5) equals the lhs of (6) = 6 but the rhs of (4)+(5) = 7 does
not. Hence the equations are inconsistent and there is no unique solution. However,
if we change the rhs of (6) to 7 then a solution is possible. Only pairs of these
equations give any information about the solution (with rhs (6) = 7).
u + v + w =2
2u
+ 3w = 5
-----------------------2v + w = 1
(4)
(5)
(5) – 2 (4) This is the equation of a line
u + v + w =2
3u + v + 4w = 6
-----------------------2v + w = 1
(4)
(6)
(6) – 3 (4) This is same the equation of a line
If instead rhs (6) = 6 then we simply have two inconsistent lines in the solution.
We can understand why by considering this example in terms of its column vectors.
1 1
 1  2
   
   
u 2   v 0   w 3    5 
 3 1
 4  6
   
   
All three vectors lie in the same plane and any one can be expressed as a linear
combination of the other two.
Exercise: prove that the three column vectors above are lie in the same plane (i.e.
they are linearly dependent).
Now consider solution of the first set of equations by Gauss elimination.
2u + v + w = 5
4u – 6v
= -2
-2u + 7v +2w = 9
(1)
(2)
(3)
In Gauss elimination we begin by subtracting multiples of the first equation from
subsequent equations so that the first column has zeros below the diagonal element.
2u + v + w = 5
– 8v -w = -12
8v +3w = 14
(1)
(2) – 2(1)
(3) –(-1)(1)
the coefficient of u is the first pivot
the coefficient of v is the second pivot
Repeating the elimination so as to eliminate the coefficient of v in the last equation
2u + v + w = 5
– 8v -w = -12
2w = 2
(1’)
(2’)
(3’) –(-1)(2’) the coefficient of w is the third pivot
Using the Gauss elimination process we have brought the equations into upper
triangular form. (If we wrote the equations in matrix form, the transformed A matrix
would have non-zero elements only on the diagonal and above). The equations can
easily be solved now using back substitution.
Gauss elimination breakdown can occur if a zero appears in a pivot position. This
might be an indication that there is no unique solution to the equations (i.e. they are
singular) or it might simply mean that we need to reorder the equations.
Non-singular example
u + v + w =
2u + 2v + 5w =
4u + 6v + 8w =
u + v + w =
3w =
2v + 4w =
(7)
(8)
(9)
(7)
(8) - 2(7)
(9) – 4(7)
reorder
u + v + w =
2v + 4w =
3w =
(7)
(9) – 4(7)
(8) - 2(7)
Singular example
u + v + w =
2u + 2v + 5w =
4u + 4v + 8w =
u + v + w=
3w =
4w =
(7)
(8)
(10)
(7)
(8) - 2(7)
(10) – 4(7)
The second pivot is unavoidably zero.
Gauss elimination by LU decomposition
In order to automate Gauss elimination for N equations in N unknowns (in matrix
form A.x = b) we need a systematic way of subtracting one row in a matrix from
another. To subtract a multiple n of row j from row i, form a square matrix L1 with
dimension N which has 1’s on the diagonal and put –n into position (i,j) and multiply
the matrix A on the left by L1.
For the equations
2u + v + w = 5
4u – 6v
= -2
-2u + 7v +2w = 9
(1)
(2)
(3)
From the previous page we know that we need to subtract 2(1) from (2) and -1(1)
from (3) so
 1 0 0


L1    2 1 0 
 0 0 1


1 0 0


L2   0 1 0 
1 0 1


Then we need to subtract -1(2’) from (3’) to reach the triangular form of the
equations.
1 0 0


L3   0 1 0 
0 1 1


Applying these operations to A in turn we have
1 1  u   2 1
1  u 
 2
u
 b 1   c1 

  
 
 
   
L 3 L 2 L 1  4  6 0  v    0  8  2  v   U v   L 3 L 2 L 1  b 2    c 2 
  2 7 2  w   0 0
 w
b  c 
1  w 

  
 
 3  3
These equations are equivalent to U.x = c. In practice
(1) Find U = L3 L2 L1.A = L-1.A
(2) Find L = (L3 L2 L1) -1= L1-1L2-1L3-1
L. U = A
(3) A.x = b = L. U. x = b => U. x = L-1. A. x = L-1. b = c
(4) Solve L.c = b for c by back substitution
(5) Solve U.x = c for x by back substitution
This is a fast method for solving simultaneous equations. It is also one of several
kinds of matrix decomposition, here A into L.U.
In situations where a zero pivot is found it may be possible to save the situation by
reordering the equations using a permutation matrix. Consider the 2x2 system
 0 2  u   b1 

    
3
4

 v   b2 
In order to perform LU decomposition we need to reorder the equations and this can
be done using a permutation matrix. It consists of the identity matrix, except that if
the ith and jth equations are to be swapped then the (i,i) and (j,j) elements become 0
and the (i,j) and (j,i) elements become 1.
0

1
3

0
1  0 2  u   0 1  b1 

   
 
0  3 4  v   1 0  b2 
4  u   b2 
    
2  v   b1 
Note that the orders of the equations and the components of b are reversed, but not the
variables. If necessary, the order of the variables can be changed by post-multiplying
by a permutation matrix. More than one permutation can be carried out using
products of permutation matrices.
Introduction to the Gnu Scientific Library (GSL)
The Reference Manual for GSL is located at www.gnu.org/software/gsl
The types of problems which GSL can handle are
Complex Numbers
Special Functions
Permutations
Sorting
Linear Algebra
Fast Fourier Transforms
Random Numbers
Random Distributions
Histograms
Monte Carlo Integration
Differential Equations
Numerical Differentiation
Series Acceleration
Root-Finding
Least-Squares Fitting
IEEE Floating-Point
Wavelets
Roots of Polynomials
Vectors and Matrices
Combinations
BLAS Support
CBLAS Library
Eigensystems
Quadrature
Quasi-Random Sequences
Statistics
N-Tuples
Simulated Annealing
Interpolation
Chebyshev Approximations
Discrete Hankel Transforms
Minimization
Physical Constants
The manual explains how to use the library in detail. We will begin by considering
use of GSL to get values of a Bessel function (Special Functions) and to solve a
system of simultaneous equations by LU decomposition. We will focus on the
Vectors and Matrices, BLAS and CBLAS, Linear Algebra and Eigensystems libraries.
All sections contain example programmes.
The GSL library is stored at /usr/local/gsl-1.2.3 on the cphys server. The header files
that you need to use a particular GSL library function are at
/usr/local/include/gsl/
and header files are included using, e.g.
#include </usr/include/gsl/gsl_sf_bessel.h>
The name gsl_sf_bessel.h means a gsl, special function call to the function Bessel
The programme bessel.c
#include <stdio.h>
#include </usr/include/gsl/gsl_sf_bessel.h>
int
main (void)
{
int i=1 ;
double x = 0.5;
double y = gsl_sf_bessel_zero_J0 (i);
double z = gsl_sf_bessel_J0 (x);
printf ("J0(%g) = %.18e\n", x, z);
return 0;
}
Calculates the first zero of the Jo Bessel function and the value of Jo(0.5). It is located
at <paste link here> and is compiled with the line
gcc -o bessel bessel.c -lgsl –lcblas
-lgsl is the gsl library and –lcblas is the C version of Basic Linear Algebra
Subroutines.
The programme lu_decomp.c
#include
#include
#include
#include
<stdio.h>
<stdlib.h>
</usr/local/include/gsl/gsl_sf_bessel.h>
</usr/local/include/gsl/gsl_linalg.h>
int
main ()
{
int s ;
double a_data[] = { 0.18,
0.41,
0.14,
0.51,
double b_data[] = { 1.00,
0.60,
0.24,
0.30,
0.13,
2.00,
0.57,
0.99,
0.97,
0.19,
3.00,
0.96,
0.58,
0.66,
0.85 };
4.00 };
gsl_matrix_view m = gsl_matrix_view_array (a_data, 4, 4);
gsl_vector_view b = gsl_vector_view_array (b_data, 4);
gsl_vector *x = gsl_vector_alloc (4);
gsl_permutation * p = gsl_permutation_alloc (4);
gsl_linalg_LU_decomp (&m.matrix, p, &s);
gsl_linalg_LU_solve (&m.matrix, p, &b.vector, x);
printf ("x = \n");
gsl_vector_fprintf (stdout, x, "%g");
gsl_permutation_free (p);
gsl_vector_free (x);
return 0;
}
Initialises the matrix m with the data in the 4x4 array a_data and the vector b with
the data in the 4 vector b_data. It allocates a solution vector x and permutation
matrix p before calculating the LU decomposition of m and solving the system of
equations
a_data.x = b_data
It then prints x before releasing the memory allocated to x and p. It compiles with
gcc -o lu_decomp lu_decomp.c -lgsl –lcblas
Exercise: Download, compile and run the programmes in Bessel.c and lu_decomp.c
Introduction to LAPACK
LAPACK is a library of linear algebra routines in fortran77 distributed with Linux or
available from netlib.org. It consists of a number of drivers, which completely solve
various problems, plus a set of computational routines which perform various specific
tasks. The naming convention for the drivers and routines is described on p 12 of the
LAPACK manual. Briefly it is: XYYZZZ where
X indicates the data type
S
single precision
D
double precision
C
complex
Z
Complex*16
YY indicates the matrix type
DI
diagonal
GB
general banded
SY
symmetric
ZZZ computation performed
Computational Routines
TRF factorise
TRS solve Ax=b
CON reciprocal condition number
RFS bounds on error
TRI find inverse
Drivers
LLS linear least squares
SEP symmetric eigenvalue problem
SVD singular value decomposition
NEP nonsymmetric eigenvalue problem
The libraries are installed on the computers in the undergraduate computer lab. We
will use C programmes to call the fortran libraries. An important difference between
C and fortran is that fortran always uses ‘call by reference’: when a subroutine is
called, the argument in the subroutine is treated as the address of the argument by the
subroutine. In C ‘call by value’ is usual for a single variable (i.e. not an array) so that
a copy of the value of the variable is passed to the function. If the value of the
variable is changed by the function and you need that value, then a call by reference
must be enforced using the argument &a instead of a for the function. (& is the
‘address of’ operator.) Also, arrays in fortran are arranged in column major order
(sequential elements in memory are arranged in columns so that A(1,3) follows
A(1,2)) whereas arrays in C are in row major order (sequential elements in memory
are in so that B[3][1] follows B[2][1]).
II Vector Spaces and Linear Equations
See Strang pp63-69
The space Rn consists of column vectors with n components which are real numbers.
e.g. R2 is the x,y plane.
1 1 0
e.g.         addition of vectors in R2
 2  0   2 
Within any vector space we can add two vectors and we can multiply them by scalars
and will obtain another vector in the same space. Conditions which vector addition
and multiplication by a scalar must meet according to the definitiuon of a vector space
are given in Strang Ex 2.1.5. The vector spaces of interest to us lie within the
standard spaces Rn. For example a 2-D plane in R3 which passes through the origin
satisfies the definition of a vector space and leads to the idea of a subspace.
Subspaces
A subspace is a non-empty subset that satisfies:
(1) Addition of two vectors in the subspace results in another vector in the subspace
(2) Multiplication by a scalar results in another vector in the subspace
The null vector, 0, forms the smallest possible subspace.
Column space of A
Consider the system of 3 equations in 2 unknowns
1 0 
 b1 
5 4 u   b 

 v   2 
2 4   b3 
For such a case where there are more equations than unknowns we expect that there
will usually be no solution. The system A.x = b can only be solved if b can be
expressed as a linear combinations of the columns of A. This can clearly be seen if
we rewrite the equations as
1 0  b1 
u 2  v 4  b2  .
3 4 b3 
A.x = b can only be solved if b lies in the plane spanned by the two column vectors.
This plane is a subspace of R3 called the column space of A denoted R(A) (the range
of A). It is a subspace of Rn where n is the number of elements in a column of A.
Null space of matrix A
Solutions to A.x = b with b = 0 always have the solution x = 0 (the null vector).
There may be additional solutions which satisfy A.xi = 0. These solutions constitute
the null space of A denoted N(A).
Solution of m equations in n unknowns
Solution for m equations in n unknowns is similar to that for n equations in n
unknowns as far as forward elimination goes. When the matrix has been reduced to
echelon form we are able to distinguish basic variables (have a pivot in echelon form)
from free variables (have no pivot in echelon form).
For a matrix with more columns n than rows m (n > m) there can be at most m pivots
and there will be at least n-m free variables. Then the homogeneous system has at
least n-m nontrivial solutions.
If the number of pivots is r then there are r basic variables and n-r free variables. r is
called the rank of the matrix.
To find homogeneous solutions (b = 0)
(1) After elimination reaches U.x = 0 identify free and basic variables.
(2) Give one free variable the value 1 and set the others to zero and solve U.x = 0 for
the basic variables.
(3) Every free variable produces its own solution and combinations of these form the
null space of A.
To find particular solutions (b ≠ 0)
(1) as above
(2) Let each free variable have the value zero
(3) solve for x
A general solution to A.x = b is a combination of particular and homogeneous
solutions.
There are 2 extreme cases for the value of the rank, r.
(1) r = n (number of columns) there are no free variables in the solution vector and
the null space contains only the null vector.
(2) r = m (number of rows) there are no zero rows in U. There are no constraints on b,
the column space is all of Rm and for every b the equations can be solved.
III Linear dependence, basis and dimension
Given a set of vectors vi look for combinations :
c1v1 + c2v2 + c3v3 + … + cnvn = 0
If there exists any solution other than ci = 0 for all coefficients ci the vectors are
linearly dependent – it is possible to express some of the vectors as linear
combinations of others.
When an n x m matrix is reduced to echelon form the non zero rows are independent
and the columns with pivots are independent.
A set of vectors in Rm must be linearly dependent if n > m.
If a vector space V consists of all linear combinations of particular vectors w1, w2, w3,
…, wn then those vectors span the space. Every vector in V can be expressed as a
combination of the wi.
To decide whether b is a combination of the columns of a matrix A composed of the
column vectors wi, solve A.x = b.
To decide whether the columns of A are independent (i.e. wi are linearly independent)
then solve A.x = 0. If a nontrivial solution can be found they are dependent.
Spanning involves the column space; independence involves the null space.
A basis for a vector space is a set of vectors : it is linearly independent and it spans
the space.
Every vector in the space is a combination of the basis vectors because they span the
space. Every combination is unique because the basis vectors are linearly
independent.
Any two bases for the same space contain the same number of vectors. This is the
dimension of the space.
IV The four fundamental subspaces
(1) The column space of A, R(A)
(2) The null space of A, N(A), contains all vectors : A.x = 0.
(3) The row space of A (the column space of AT), R(AT)
(4) The left null space of A, N(AT), contains all vectors : y . AT = 0
The row space of A R(AT) has the same dimension r as the row space of U and it has
the same bases because the two row spaces are the same.
The null space of A N(A) is the same as the null space of U. If the system A.x = 0 is
reduced to U.x = 0 then none of the solutions (which constitute the null space) is
changed. It has dimension n – r. If homogeneous solutions to U.x = 0 are found, they
constitute a basis for N(A).
The column space or range of A R(A). We find a basis for R(A) by finding a basis
for R(U). This latter basis is found by identifying the columns of U which contain
pivots – all other columns of U can be expressed in terms of these pivot column
vectors. The columns of A which form a basis for R(A) are the same columns as
those which form a basis for U because both A.x = 0 and U.x = 0 share the same
inhomogeneous solutions, x. Combinations of columns of A or U with coefficients
given by the components of x for a basis for either A or U.
The left nullspace of A N(AT). AT is an n x m matrix and its null space is a subspace
of Rm. The dimension of the column space of a matrix plus the dimension of the null
space must add to equal the number of columns. Since AT has m columns and the
dimension of the column space of AT is r the left null space must have dimension m –
r.
Fundamental theorem of linear algebra
R(A) = column space of A
N(A) = null space of A
R(AT) = row space of A
N(AT) = left nullspace of A
dimension r
dimension n – r
dimension r
dimension m- r
Inverses and transposes
Inverse of a square matrix is defined by A.A-1 = I where I is the nxn unit matrix.
The inverse of a product of matrices is given by (AB) -1 = B-1 A-1 (prove by pre and
post-multiplying by AB.
One test for invertibility is that a matrix should have a full set of nonzero pivots. If so,
by definition, it is non-singular and is therefore invertible. A square matrix is
invertible iff it is non-singular.
Transposes of matrices are defined below:
(AT)ij = A ji
(A+B) T = AT + B T
(A B) T = B T AT
(A-1) T = (AT) -1
If AT = A A is a symmetric matrix
If A-1exists it is also symmetric
If A is symmetric and can be factored into LDU without row or column exchanges
which would destroy the symmetry the U = LT and A = LDLT is the LDU
factorisation.
Problem 1. Use the programme dgetri.c to invert a matrix by solving A-1 L = U-1 for
A-1. DGETRI requires P, L and U which are computed by first calling DGETRF.
The output should pre and post-multiply the inverse by the original matrix to show
that the inverse is indeed obtained.
The programme dgetri.c calls the LAPACK routines DGETRF and DGETRI to
factorise a square matrix into LU form and then invert it by Gauss-Jordan elimination.
They are available at
www.tcd.ie/Physics/People/Charles.Patterson/SS/SS4062
and dgetri.c is compiled with cc –o dgetri –llapack –lblas –lg2c dgetri.c
–lblas is a call to the basic linear algebra subroutines which come in 3 levels:
Level 1
Level2
Level 3
vector operations
matrix-vector operations
matrix-matrix operations
y = ax + y
y = aAx + by
C = aAB + bC
-lg2c is the Gnu fortran to c converter. Information on how to call the DGETRF and
DGETRI routines can be found from the LAPACK man pages >man dgetrf
> man dgetri