Computes the condition number of a matrix.
Function Return Value
Computes condition number of matrix A. This is a scalar for the case where A is rank-2 or a sparse matrix. It is a rank-1 array when A is a dense rank-3 array. (Output)
Required Argument
A —
Matrix for which the condition number is to be computed. The
matrix may be real, double, complex, double-complex, or one of the
computational sparse matrix derived types, ?_hbc_sparse.
For an array of type real, double, complex, or double-complex the array may be
of rank-2 or rank-3.
For a dense rank-3 array, each rank-2 array section,
(for fixed third subscript), is a separate problem. In this case, the output is
a rank-1 array of condition numbers for each problem. (Input)
Optional Arguments, Packaged Options
NORM_CHOICE — Integer indicating the type of norm to be used in computing the condition number.
|
NORM_CHOICE |
CONDITION |
Square Matrix |
Rectangular Matrix | ||
|
Dense |
Sparse |
Dense |
Sparse | ||
|
1 |
|
Yes |
Yes |
No |
No |
|
2 (Default) |
|
Yes |
Yes |
Yes |
No |
|
huge(1) |
|
Yes |
Yes |
No |
No |
This function uses lin_sol_svd (see Chapter 1, “Linear Systems”).
The option and derived type names are given in the following tables:
|
Option Names for COND |
Option Value |
|
?_cond_set_small |
1 |
|
?_cond_for_lin_sol_svd |
2 |
|
Name of
Unallocated Option Array |
Use |
Derived Type |
|
?_cond_options(:) |
Use when setting options for calls hereafter. |
?_options |
|
?_cond_options_once(:) |
Use when setting options for next call only. |
?_options |
For a description on how to use these options, see Matrix Optional Data Changes. See lin_sol_svd located in Chapter 1, “Linear Systems” for the specific options for this routine.
FORTRAN 90 Interface
COND (A [,…])
Description
The mathematical definitions of the condition numbers which this routine estimates are:
COND can be used with either dense or sparse matrices as follows:
|
Square Matrix |
Rectangular Matrix | |||
|
|
Dense |
Sparse |
Dense |
Sparse |
|
|
Yes |
Yes |
No |
No |
|
|
Yes |
Yes |
Yes |
No |
|
|
Yes |
Yes |
No |
No |
The generic function COND
can be used with either dense or sparse square matrices. This function
uses lin_sol_svd
for dense square and rectangular matrices in computing 
.
The function uses lin_sol_gen for
dense square matrices in computing 
and
. For sparse square matrices, the values returned
forandare provided by the SuperLU linear
equation solver. The condition numberis computed by an algorithm that
first approximates
by
computing the singular values of the
bidiagonal matrix obtained using the Lanczos method found in
Golub and Van Loan, Ed. 3, p. 495. Here 
is set using the value A%Options%Cond_Iteration_Max,
which has the default value of 30. The value
is obtained using the power
method, Golub and Van Loan, p. 330, iterating with the inverse matrix
. For complex
matrices
is replaced
by
. The dominant
eigenvalue of this inverse matrix is. The number of iterations is limited by the parameter
value or relative
accuracy equal to the cube root of machine epsilon. Some timing tests
indicate that computing 
for sparse matrices by this algorithm typically requires about
twice the time as for a single linear solve using the defined operator A .ix.
b.
For computation ofwith rectangular sparse matrices one can use a dense matrix
representation for the matrix. This is not recommended except for small
problem sizes. For overdetermined systems of sparse
least-squares equations
a related square system is given by
One can form 
, which has more than twice the number of non-zeros as
. But is still sparse. One can
use the condition number ofas an estimate of the accuracy for the solution vector
and the residual vector 
. Note that this
version of the condition number is not the same as the
condition number of but is relevant to determining the
accuracy of the least-squares system.
Examples
Dense Matrix Example (operator_ex02.f90)
use wrrrn_int
use linear_operators
integer, parameter :: N=3
real (kind(1.e0)) A(N,N)
real (kind(1.e0)) C1, C2, CINF
DATA A/2.0, 2.0, -4.0, 0.0, -1.0, 2.0, 0.0, 0.0, 5.0/
CINF = COND (A, norm_choice=huge(1))
C1 = COND (A, norm_choice=1)
C2 = COND (A)
call wrrrn ( 'A', A)
write (*,*) 'L1 condition number= ', C1
write (*,*) 'L2 condition number= ', C2
write (*,*) 'L infinity condition number= ', CINF
end
Output
A
1 2 3
1 2.000 0.000 0.000
2 2.000 -1.000 0.000
3 -4.000 2.000 5.000
L1 condition number= 12.0
L2 condition number= 10.405088
L infinity condition number= 22.0
Sparse Matrix Example
use wrrrn_int
use linear_operators
type (s_sparse) S
type (s_hbc_sparse) H
integer, parameter :: N=3
real (kind(1.e0)) X(N,N)
real (kind(1.e0)) C1, C2, CINF
S = s_entry (1, 1, 2.0)
S = s_entry (2, 1, 2.0)
S = s_entry (3, 1, -4.0)
S = s_entry (3, 2, 2.0)
S = s_entry (2, 2, -1.0)
S = s_entry (3, 3, 5.0)
H = S ! sparse
X = H ! dense equivalent of H
CINF = COND (H, norm_choice=huge(1))
C1 = COND (H, norm_choice=1)
C2 = COND (H)
call wrrrn ( 'H', X)
write (*,*) 'L1 condition number= ', C1
write (*,*) 'L2 condition number= ', C2
write (*,*) 'L infinity condition number= ', CINF
end
Output
H
1 2 3
1 2.000 0.000 0.000
2 2.000 -1.000 0.000
3 -4.000 2.000 5.000
L1 condition number= 12.0
L2 condition number= 10.405088
L infinity condition number= 22.0
Parallel Example (parallel_ex02.f90)
use linear_operators
use mpi_setup_int
implicit none
! This is the equivalent of Parallel Example 2 for .i. and det() with box
! data types, operators and functions.
integer, parameter :: n=32, nr=4
integer J
real(kind(1e0)) :: one=1e0
real(kind(1e0)), dimension(nr) :: err, det_A, det_i
real(kind(1e0)), dimension(n,n,nr) :: A, inv, R, S
! Setup for MPI.
MP_NPROCS=MP_SETUP()
! Generate a random matrix.
A = rand(A)
! Compute the matrix inverse and its determinant.
inv = .i.A; det_A = det(A)
! Compute the determinant for the inverse matrix.
det_i = det(inv)
! Check the quality of both left and right inverses.
DO J=1,nr; R(:,:,J)=EYE(N); END DO
S=R; R=R-(A .x. inv); S=S-(inv .x. A)
err = (norm(R)+norm(S))/cond(A)
if (ALL(err <= sqrt(epsilon(one)) .and. &
abs(det_A*det_i - one) <= sqrt(epsilon(one)))&
.and. MP_RANK == 0) &
write (*,*) 'Parallel Example 2 is correct.'
! See to any error messages and quit MPI.
MP_NPROCS=MP_SETUP('Final')
end
|
Visual Numerics, Inc. PHONE: 713.784.3131 FAX:713.781.9260 |