CLATMT
Craig Lucas, University of Manchester / NAG Ltd.
October, 2008
October, 2008
Purpose
CLATMT generates random matrices with specified singular values
(or hermitian with specified eigenvalues)
for testing LAPACK programs.
CLATMT operates by applying the following sequence of
operations:
Set the diagonal to D, where D may be input or
computed according to MODE, COND, DMAX, and SYM
as described below.
Generate a matrix with the appropriate band structure, by one
of two methods:
Method A:
Generate a dense M x N matrix by multiplying D on the left
and the right by random unitary matrices, then:
Reduce the bandwidth according to KL and KU, using
Householder transformations.
Method B:
Convert the bandwidth-0 (i.e., diagonal) matrix to a
bandwidth-1 matrix using Givens rotations, "chasing"
out-of-band elements back, much as in QR; then convert
the bandwidth-1 to a bandwidth-2 matrix, etc. Note
that for reasonably small bandwidths (relative to M and
N) this requires less storage, as a dense matrix is not
generated. Also, for hermitian or symmetric matrices,
only one triangle is generated.
Method A is chosen if the bandwidth is a large fraction of the
order of the matrix, and LDA is at least M (so a dense
matrix can be stored.) Method B is chosen if the bandwidth
is small (< 1/2 N for hermitian or symmetric, < .3 N+M for
non-symmetric), or LDA is less than M and not less than the
bandwidth.
Pack the matrix if desired. Options specified by PACK are:
no packing
zero out upper half (if hermitian)
zero out lower half (if hermitian)
store the upper half columnwise (if hermitian or upper
triangular)
store the lower half columnwise (if hermitian or lower
triangular)
store the lower triangle in banded format (if hermitian or
lower triangular)
store the upper triangle in banded format (if hermitian or
upper triangular)
store the entire matrix in banded format
If Method B is chosen, and band format is specified, then the
matrix will be generated in the band format, so no repacking
will be necessary.
(or hermitian with specified eigenvalues)
for testing LAPACK programs.
CLATMT operates by applying the following sequence of
operations:
Set the diagonal to D, where D may be input or
computed according to MODE, COND, DMAX, and SYM
as described below.
Generate a matrix with the appropriate band structure, by one
of two methods:
Method A:
Generate a dense M x N matrix by multiplying D on the left
and the right by random unitary matrices, then:
Reduce the bandwidth according to KL and KU, using
Householder transformations.
Method B:
Convert the bandwidth-0 (i.e., diagonal) matrix to a
bandwidth-1 matrix using Givens rotations, "chasing"
out-of-band elements back, much as in QR; then convert
the bandwidth-1 to a bandwidth-2 matrix, etc. Note
that for reasonably small bandwidths (relative to M and
N) this requires less storage, as a dense matrix is not
generated. Also, for hermitian or symmetric matrices,
only one triangle is generated.
Method A is chosen if the bandwidth is a large fraction of the
order of the matrix, and LDA is at least M (so a dense
matrix can be stored.) Method B is chosen if the bandwidth
is small (< 1/2 N for hermitian or symmetric, < .3 N+M for
non-symmetric), or LDA is less than M and not less than the
bandwidth.
Pack the matrix if desired. Options specified by PACK are:
no packing
zero out upper half (if hermitian)
zero out lower half (if hermitian)
store the upper half columnwise (if hermitian or upper
triangular)
store the lower half columnwise (if hermitian or lower
triangular)
store the lower triangle in banded format (if hermitian or
lower triangular)
store the upper triangle in banded format (if hermitian or
upper triangular)
store the entire matrix in banded format
If Method B is chosen, and band format is specified, then the
matrix will be generated in the band format, so no repacking
will be necessary.
Arguments
| M |
(input) INTEGER
The number of rows of A. Not modified.
|
| N |
(input) INTEGER
The number of columns of A. N must equal M if the matrix
is symmetric or hermitian (i.e., if SYM is not 'N') Not modified. |
| DIST |
(input) CHARACTER*1
On entry, DIST specifies the type of distribution to be used
to generate the random eigen-/singular values. 'U' => UNIFORM( 0, 1 ) ( 'U' for uniform ) 'S' => UNIFORM( -1, 1 ) ( 'S' for symmetric ) 'N' => NORMAL( 0, 1 ) ( 'N' for normal ) Not modified. |
| ISEED |
(input/output) INTEGER array, dimension ( 4 )
On entry ISEED specifies the seed of the random number
generator. They should lie between 0 and 4095 inclusive, and ISEED(4) should be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers. The values of ISEED are changed on exit, and can be used in the next call to CLATMT to continue the same random number sequence. Changed on exit. |
| SYM |
(input) CHARACTER*1
If SYM='H', the generated matrix is hermitian, with
eigenvalues specified by D, COND, MODE, and DMAX; they may be positive, negative, or zero. If SYM='P', the generated matrix is hermitian, with eigenvalues (= singular values) specified by D, COND, MODE, and DMAX; they will not be negative. If SYM='N', the generated matrix is nonsymmetric, with singular values specified by D, COND, MODE, and DMAX; they will not be negative. If SYM='S', the generated matrix is (complex) symmetric, with singular values specified by D, COND, MODE, and DMAX; they will not be negative. Not modified. |
| D |
(input/output) REAL array, dimension ( MIN( M, N ) )
This array is used to specify the singular values or
eigenvalues of A (see SYM, above.) If MODE=0, then D is assumed to contain the singular/eigenvalues, otherwise they will be computed according to MODE, COND, and DMAX, and placed in D. Modified if MODE is nonzero. |
| MODE |
(input) INTEGER
On entry this describes how the singular/eigenvalues are to
be specified: MODE = 0 means use D as input MODE = 1 sets D(1)=1 and D(2:RANK)=1.0/COND MODE = 2 sets D(1:RANK-1)=1 and D(RANK)=1.0/COND MODE = 3 sets D(I)=COND**(-(I-1)/(RANK-1)) MODE = 4 sets D(i)=1 - (i-1)/(N-1)*(1 - 1/COND) MODE = 5 sets D to random numbers in the range ( 1/COND , 1 ) such that their logarithms are uniformly distributed. MODE = 6 set D to random numbers from same distribution as the rest of the matrix. MODE < 0 has the same meaning as ABS(MODE), except that the order of the elements of D is reversed. Thus if MODE is positive, D has entries ranging from 1 to 1/COND, if negative, from 1/COND to 1, If SYM='H', and MODE is neither 0, 6, nor -6, then the elements of D will also be multiplied by a random sign (i.e., +1 or -1.) Not modified. |
| COND |
(input) REAL
On entry, this is used as described under MODE above.
If used, it must be >= 1. Not modified. |
| DMAX |
(input) REAL
If MODE is neither -6, 0 nor 6, the contents of D, as
computed according to MODE and COND, will be scaled by DMAX / max(abs(D(i))); thus, the maximum absolute eigen- or singular value (which is to say the norm) will be abs(DMAX). Note that DMAX need not be positive: if DMAX is negative (or zero), D will be scaled by a negative number (or zero). Not modified. |
| RANK |
(input) INTEGER
The rank of matrix to be generated for modes 1,2,3 only.
D( RANK+1:N ) = 0. Not modified. |
| KL |
(input) INTEGER
This specifies the lower bandwidth of the matrix. For
example, KL=0 implies upper triangular, KL=1 implies upper Hessenberg, and KL being at least M-1 means that the matrix has full lower bandwidth. KL must equal KU if the matrix is symmetric or hermitian. Not modified. |
| KU |
(input) INTEGER
This specifies the upper bandwidth of the matrix. For
example, KU=0 implies lower triangular, KU=1 implies lower Hessenberg, and KU being at least N-1 means that the matrix has full upper bandwidth. KL must equal KU if the matrix is symmetric or hermitian. Not modified. |
| PACK |
(input) CHARACTER*1
This specifies packing of matrix as follows:
'N' => no packing 'U' => zero out all subdiagonal entries (if symmetric or hermitian) 'L' => zero out all superdiagonal entries (if symmetric or hermitian) 'C' => store the upper triangle columnwise (only if the matrix is symmetric, hermitian, or upper triangular) 'R' => store the lower triangle columnwise (only if the matrix is symmetric, hermitian, or lower triangular) 'B' => store the lower triangle in band storage scheme (only if the matrix is symmetric, hermitian, or lower triangular) 'Q' => store the upper triangle in band storage scheme (only if the matrix is symmetric, hermitian, or upper triangular) 'Z' => store the entire matrix in band storage scheme (pivoting can be provided for by using this option to store A in the trailing rows of the allocated storage) Using these options, the various LAPACK packed and banded storage schemes can be obtained: GB - use 'Z' PB, SB, HB, or TB - use 'B' or 'Q' PP, SP, HB, or TP - use 'C' or 'R' If two calls to CLATMT differ only in the PACK parameter, they will generate mathematically equivalent matrices. Not modified. |
| A |
(input/output) COMPLEX array, dimension ( LDA, N )
On exit A is the desired test matrix. A is first generated
in full (unpacked) form, and then packed, if so specified by PACK. Thus, the first M elements of the first N columns will always be modified. If PACK specifies a packed or banded storage scheme, all LDA elements of the first N columns will be modified; the elements of the array which do not correspond to elements of the generated matrix are set to zero. Modified. |
| LDA |
(input) INTEGER
LDA specifies the first dimension of A as declared in the
calling program. If PACK='N', 'U', 'L', 'C', or 'R', then LDA must be at least M. If PACK='B' or 'Q', then LDA must be at least MIN( KL, M-1) (which is equal to MIN(KU,N-1)). If PACK='Z', LDA must be large enough to hold the packed array: MIN( KU, N-1) + MIN( KL, M-1) + 1. Not modified. |
| WORK |
(workspace) COMPLEX array, dimension ( 3*MAX( N, M ) )
Workspace.
Modified. |
| INFO |
(output) INTEGER
Error code. On exit, INFO will be set to one of the
following values: 0 => normal return -1 => M negative or unequal to N and SYM='S', 'H', or 'P' -2 => N negative -3 => DIST illegal string -5 => SYM illegal string -7 => MODE not in range -6 to 6 -8 => COND less than 1.0, and MODE neither -6, 0 nor 6 -10 => KL negative -11 => KU negative, or SYM is not 'N' and KU is not equal to KL -12 => PACK illegal string, or PACK='U' or 'L', and SYM='N'; or PACK='C' or 'Q' and SYM='N' and KL is not zero; or PACK='R' or 'B' and SYM='N' and KU is not zero; or PACK='U', 'L', 'C', 'R', 'B', or 'Q', and M is not N. -14 => LDA is less than M, or PACK='Z' and LDA is less than MIN(KU,N-1) + MIN(KL,M-1) + 1. 1 => Error return from SLATM7 2 => Cannot scale to DMAX (max. sing. value is 0) 3 => Error return from CLAGGE, CLAGHE or CLAGSY |