SLATMR
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
June 2010
June 2010
Purpose
SLATMR generates random matrices of various types for testing
LAPACK programs.
SLATMR operates by applying the following sequence of
operations:
Generate a matrix A with random entries of distribution DIST
which is symmetric if SYM='S', and nonsymmetric
if SYM='N'.
Set the diagonal to D, where D may be input or
computed according to MODE, COND, DMAX and RSIGN
as described below.
Grade the matrix, if desired, from the left and/or right
as specified by GRADE. The inputs DL, MODEL, CONDL, DR,
MODER and CONDR also determine the grading as described
below.
Permute, if desired, the rows and/or columns as specified by
PIVTNG and IPIVOT.
Set random entries to zero, if desired, to get a random sparse
matrix as specified by SPARSE.
Make A a band matrix, if desired, by zeroing out the matrix
outside a band of lower bandwidth KL and upper bandwidth KU.
Scale A, if desired, to have maximum entry ANORM.
Pack the matrix if desired. Options specified by PACK are:
no packing
zero out upper half (if symmetric)
zero out lower half (if symmetric)
store the upper half columnwise (if symmetric or
square upper triangular)
store the lower half columnwise (if symmetric or
square lower triangular)
same as upper half rowwise if symmetric
store the lower triangle in banded format (if symmetric)
store the upper triangle in banded format (if symmetric)
store the entire matrix in banded format
Note: If two calls to SLATMR differ only in the PACK parameter,
they will generate mathematically equivalent matrices.
If two calls to SLATMR both have full bandwidth (KL = M-1
and KU = N-1), and differ only in the PIVTNG and PACK
parameters, then the matrices generated will differ only
in the order of the rows and/or columns, and otherwise
contain the same data. This consistency cannot be and
is not maintained with less than full bandwidth.
LAPACK programs.
SLATMR operates by applying the following sequence of
operations:
Generate a matrix A with random entries of distribution DIST
which is symmetric if SYM='S', and nonsymmetric
if SYM='N'.
Set the diagonal to D, where D may be input or
computed according to MODE, COND, DMAX and RSIGN
as described below.
Grade the matrix, if desired, from the left and/or right
as specified by GRADE. The inputs DL, MODEL, CONDL, DR,
MODER and CONDR also determine the grading as described
below.
Permute, if desired, the rows and/or columns as specified by
PIVTNG and IPIVOT.
Set random entries to zero, if desired, to get a random sparse
matrix as specified by SPARSE.
Make A a band matrix, if desired, by zeroing out the matrix
outside a band of lower bandwidth KL and upper bandwidth KU.
Scale A, if desired, to have maximum entry ANORM.
Pack the matrix if desired. Options specified by PACK are:
no packing
zero out upper half (if symmetric)
zero out lower half (if symmetric)
store the upper half columnwise (if symmetric or
square upper triangular)
store the lower half columnwise (if symmetric or
square lower triangular)
same as upper half rowwise if symmetric
store the lower triangle in banded format (if symmetric)
store the upper triangle in banded format (if symmetric)
store the entire matrix in banded format
Note: If two calls to SLATMR differ only in the PACK parameter,
they will generate mathematically equivalent matrices.
If two calls to SLATMR both have full bandwidth (KL = M-1
and KU = N-1), and differ only in the PIVTNG and PACK
parameters, then the matrices generated will differ only
in the order of the rows and/or columns, and otherwise
contain the same data. This consistency cannot be and
is not maintained with less than full bandwidth.
Arguments
| M |
(input) INTEGER
Number of rows of A. Not modified.
|
| N |
(input) INTEGER
Number of columns of A. Not modified.
|
| DIST |
(input) CHARACTER*1
On entry, DIST specifies the type of distribution to be used
to generate a random matrix . '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 SLATMR to continue the same random number sequence. Changed on exit. |
| SYM |
(input) CHARACTER*1
If SYM='S' or 'H', generated matrix is symmetric.
If SYM='N', generated matrix is nonsymmetric. Not modified. |
| D |
(input) REAL array, dimension (min(M,N))
On entry this array specifies the diagonal entries
of the diagonal of A. D may either be specified on entry, or set according to MODE and COND as described below. May be changed on exit if MODE is nonzero. |
| MODE |
(input) INTEGER
On entry describes how D is to be used:
MODE = 0 means use D as input MODE = 1 sets D(1)=1 and D(2:N)=1.0/COND MODE = 2 sets D(1:N-1)=1 and D(N)=1.0/COND MODE = 3 sets D(I)=COND**(-(I-1)/(N-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, Not modified. |
| COND |
(input) REAL
On entry, used as described under MODE above.
If used, it must be >= 1. Not modified. |
| DMAX |
(input) REAL
If MODE neither -6, 0 nor 6, the diagonal is scaled by
DMAX / max(abs(D(i))), so that maximum absolute entry of diagonal is abs(DMAX). If DMAX is negative (or zero), diagonal will be scaled by a negative number (or zero). |
| RSIGN |
(input) CHARACTER*1
If MODE neither -6, 0 nor 6, specifies sign of diagonal
as follows: 'T' => diagonal entries are multiplied by 1 or -1 with probability .5 'F' => diagonal unchanged Not modified. |
| GRADE |
(input) CHARACTER*1
Specifies grading of matrix as follows:
'N' => no grading 'L' => matrix premultiplied by diag( DL ) (only if matrix nonsymmetric) 'R' => matrix postmultiplied by diag( DR ) (only if matrix nonsymmetric) 'B' => matrix premultiplied by diag( DL ) and postmultiplied by diag( DR ) (only if matrix nonsymmetric) 'S' or 'H' => matrix premultiplied by diag( DL ) and postmultiplied by diag( DL ) ('S' for symmetric, or 'H' for Hermitian) 'E' => matrix premultiplied by diag( DL ) and postmultiplied by inv( diag( DL ) ) ( 'E' for eigenvalue invariance) (only if matrix nonsymmetric) Note: if GRADE='E', then M must equal N. Not modified. |
| DL |
(input/output) REAL array, dimension (M)
If MODEL=0, then on entry this array specifies the diagonal
entries of a diagonal matrix used as described under GRADE above. If MODEL is not zero, then DL will be set according to MODEL and CONDL, analogous to the way D is set according to MODE and COND (except there is no DMAX parameter for DL). If GRADE='E', then DL cannot have zero entries. Not referenced if GRADE = 'N' or 'R'. Changed on exit. |
| MODEL |
(input) INTEGER
This specifies how the diagonal array DL is to be computed,
just as MODE specifies how D is to be computed. Not modified. |
| CONDL |
(input) REAL
When MODEL is not zero, this specifies the condition number
of the computed DL. Not modified. |
| DR |
(input/output) REAL array, dimension (N)
If MODER=0, then on entry this array specifies the diagonal
entries of a diagonal matrix used as described under GRADE above. If MODER is not zero, then DR will be set according to MODER and CONDR, analogous to the way D is set according to MODE and COND (except there is no DMAX parameter for DR). Not referenced if GRADE = 'N', 'L', 'H', 'S' or 'E'. Changed on exit. |
| MODER |
(input) INTEGER
This specifies how the diagonal array DR is to be computed,
just as MODE specifies how D is to be computed. Not modified. |
| CONDR |
(input) REAL
When MODER is not zero, this specifies the condition number
of the computed DR. Not modified. |
| PIVTNG |
(input) CHARACTER*1
On entry specifies pivoting permutations as follows:
'N' or ' ' => none. 'L' => left or row pivoting (matrix must be nonsymmetric). 'R' => right or column pivoting (matrix must be nonsymmetric). 'B' or 'F' => both or full pivoting, i.e., on both sides. In this case, M must equal N If two calls to SLATMR both have full bandwidth (KL = M-1 and KU = N-1), and differ only in the PIVTNG and PACK parameters, then the matrices generated will differ only in the order of the rows and/or columns, and otherwise contain the same data. This consistency cannot be maintained with less than full bandwidth. |
| IPIVOT |
(input) INTEGER array, dimension (N or M)
This array specifies the permutation used. After the
basic matrix is generated, the rows, columns, or both are permuted. If, say, row pivoting is selected, SLATMR starts with the *last* row and interchanges the M-th and IPIVOT(M)-th rows, then moves to the next-to-last row, interchanging the (M-1)-th and the IPIVOT(M-1)-th rows, and so on. In terms of "2-cycles", the permutation is (1 IPIVOT(1)) (2 IPIVOT(2)) ... (M IPIVOT(M)) where the rightmost cycle is applied first. This is the *inverse* of the effect of pivoting in LINPACK. The idea is that factoring (with pivoting) an identity matrix which has been inverse-pivoted in this way should result in a pivot vector identical to IPIVOT. Not referenced if PIVTNG = 'N'. Not modified. |
| SPARSE |
(input) REAL
On entry specifies the sparsity of the matrix if a sparse
matrix is to be generated. SPARSE should lie between 0 and 1. To generate a sparse matrix, for each matrix entry a uniform ( 0, 1 ) random number x is generated and compared to SPARSE; if x is larger the matrix entry is unchanged and if x is smaller the entry is set to zero. Thus on the average a fraction SPARSE of the entries will be set to zero. Not modified. |
| KL |
(input) INTEGER
On entry specifies the lower bandwidth of the matrix. For
example, KL=0 implies upper triangular, KL=1 implies upper Hessenberg, and KL at least M-1 implies the matrix is not banded. Must equal KU if matrix is symmetric. Not modified. |
| KU |
(input) INTEGER
On entry specifies the upper bandwidth of the matrix. For
example, KU=0 implies lower triangular, KU=1 implies lower Hessenberg, and KU at least N-1 implies the matrix is not banded. Must equal KL if matrix is symmetric. Not modified. |
| ANORM |
(input) REAL
On entry specifies maximum entry of output matrix
(output matrix will by multiplied by a constant so that its largest absolute entry equal ANORM) if ANORM is nonnegative. If ANORM is negative no scaling is done. Not modified. |
| PACK |
(input) CHARACTER*1
On entry specifies packing of matrix as follows:
'N' => no packing 'U' => zero out all subdiagonal entries (if symmetric) 'L' => zero out all superdiagonal entries (if symmetric) 'C' => store the upper triangle columnwise (only if matrix symmetric or square upper triangular) 'R' => store the lower triangle columnwise (only if matrix symmetric or square lower triangular) (same as upper half rowwise if symmetric) 'B' => store the lower triangle in band storage scheme (only if matrix symmetric) 'Q' => store the upper triangle in band storage scheme (only if matrix symmetric) '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 or TB - use 'B' or 'Q' PP, SP or TP - use 'C' or 'R' If two calls to SLATMR differ only in the PACK parameter, they will generate mathematically equivalent matrices. Not modified. |
| A |
(input/output) REAL array, dimension (LDA,N)
On exit A is the desired test matrix. Only those
entries of A which are significant on output will be referenced (even if A is in packed or band storage format). The 'unoccupied corners' of A in band format will be zeroed out. |
| LDA |
(input) INTEGER
on entry LDA specifies the first dimension of A as
declared in the calling program. If PACK='N', 'U' or 'L', LDA must be at least max ( 1, M ). If PACK='C' or 'R', LDA must be at least 1. If PACK='B', or 'Q', LDA must be MIN ( KU+1, N ) If PACK='Z', LDA must be at least KUU+KLL+1, where KUU = MIN ( KU, N-1 ) and KLL = MIN ( KL, N-1 ) Not modified. |
| IWORK |
(workspace) INTEGER array, dimension ( N or M)
Workspace. Not referenced if PIVTNG = 'N'. Changed on exit.
|
| INFO |
(output) INTEGER
Error parameter on exit:
0 => normal return -1 => M negative or unequal to N and SYM='S' or 'H' -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 => MODE neither -6, 0 nor 6 and RSIGN illegal string -11 => GRADE illegal string, or GRADE='E' and M not equal to N, or GRADE='L', 'R', 'B' or 'E' and SYM = 'S' or 'H' -12 => GRADE = 'E' and DL contains zero -13 => MODEL not in range -6 to 6 and GRADE= 'L', 'B', 'H', 'S' or 'E' -14 => CONDL less than 1.0, GRADE='L', 'B', 'H', 'S' or 'E', and MODEL neither -6, 0 nor 6 -16 => MODER not in range -6 to 6 and GRADE= 'R' or 'B' -17 => CONDR less than 1.0, GRADE='R' or 'B', and MODER neither -6, 0 nor 6 -18 => PIVTNG illegal string, or PIVTNG='B' or 'F' and M not equal to N, or PIVTNG='L' or 'R' and SYM='S' or 'H' -19 => IPIVOT contains out of range number and PIVTNG not equal to 'N' -20 => KL negative -21 => KU negative, or SYM='S' or 'H' and KU not equal to KL -22 => SPARSE not in range 0. to 1. -24 => PACK illegal string, or PACK='U', 'L', 'B' or 'Q' and SYM='N', or PACK='C' and SYM='N' and either KL not equal to 0 or N not equal to M, or PACK='R' and SYM='N', and either KU not equal to 0 or N not equal to M -26 => LDA too small 1 => Error return from SLATM1 (computing D) 2 => Cannot scale diagonal to DMAX (max. entry is 0) 3 => Error return from SLATM1 (computing DL) 4 => Error return from SLATM1 (computing DR) 5 => ANORM is positive, but matrix constructed prior to attempting to scale it to have norm ANORM, is zero |