Provided by: liblapack-doc-man_3.6.0-2ubuntu2_all 
NAME
real_matgen - real
Functions
subroutine slagge (M, N, KL, KU, D, A, LDA, ISEED, WORK, INFO)
SLAGGE
subroutine slagsy (N, K, D, A, LDA, ISEED, WORK, INFO)
SLAGSY
subroutine slahilb (N, NRHS, A, LDA, X, LDX, B, LDB, WORK, INFO)
SLAHILB
subroutine slakf2 (M, N, A, LDA, B, D, E, Z, LDZ)
SLAKF2
real function slaran (ISEED)
SLARAN
subroutine slarge (N, A, LDA, ISEED, WORK, INFO)
SLARGE
real function slarnd (IDIST, ISEED)
SLARND
subroutine slaror (SIDE, INIT, M, N, A, LDA, ISEED, X, INFO)
SLAROR
subroutine slarot (LROWS, LLEFT, LRIGHT, NL, C, S, A, LDA, XLEFT, XRIGHT)
SLAROT
subroutine slatm1 (MODE, COND, IRSIGN, IDIST, ISEED, D, N, INFO)
SLATM1
real function slatm2 (M, N, I, J, KL, KU, IDIST, ISEED, D, IGRADE, DL, DR, IPVTNG, IWORK,
SPARSE)
SLATM2
real function slatm3 (M, N, I, J, ISUB, JSUB, KL, KU, IDIST, ISEED, D, IGRADE, DL, DR,
IPVTNG, IWORK, SPARSE)
SLATM3
subroutine slatm5 (PRTYPE, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, R, LDR,
L, LDL, ALPHA, QBLCKA, QBLCKB)
SLATM5
subroutine slatm6 (TYPE, N, A, LDA, B, X, LDX, Y, LDY, ALPHA, BETA, WX, WY, S, DIF)
SLATM6
subroutine slatm7 (MODE, COND, IRSIGN, IDIST, ISEED, D, N, RANK, INFO)
SLATM7
subroutine slatme (N, DIST, ISEED, D, MODE, COND, DMAX, EI, RSIGN, UPPER, SIM, DS, MODES,
CONDS, KL, KU, ANORM, A, LDA, WORK, INFO)
SLATME
subroutine slatmr (M, N, DIST, ISEED, SYM, D, MODE, COND, DMAX, RSIGN, GRADE, DL, MODEL,
CONDL, DR, MODER, CONDR, PIVTNG, IPIVOT, KL, KU, SPARSE, ANORM, PACK, A, LDA, IWORK,
INFO)
SLATMR
subroutine slatms (M, N, DIST, ISEED, SYM, D, MODE, COND, DMAX, KL, KU, PACK, A, LDA,
WORK, INFO)
SLATMS
subroutine slatmt (M, N, DIST, ISEED, SYM, D, MODE, COND, DMAX, RANK, KL, KU, PACK, A,
LDA, WORK, INFO)
SLATMT
Detailed Description
This is the group of real LAPACK TESTING MATGEN routines.
Function Documentation
subroutine slagge (integer M, integer N, integer KL, integer KU, real, dimension( * ) D, real,
dimension( lda, * ) A, integer LDA, integer, dimension( 4 ) ISEED, real, dimension( * )
WORK, integer INFO)
SLAGGE
Purpose:
SLAGGE generates a real general m by n matrix A, by pre- and post-
multiplying a real diagonal matrix D with random orthogonal matrices:
A = U*D*V. The lower and upper bandwidths may then be reduced to
kl and ku by additional orthogonal transformations.
Parameters:
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
KL
KL is INTEGER
The number of nonzero subdiagonals within the band of A.
0 <= KL <= M-1.
KU
KU is INTEGER
The number of nonzero superdiagonals within the band of A.
0 <= KU <= N-1.
D
D is REAL array, dimension (min(M,N))
The diagonal elements of the diagonal matrix D.
A
A is REAL array, dimension (LDA,N)
The generated m by n matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= M.
ISEED
ISEED is INTEGER array, dimension (4)
On entry, the seed of the random number generator; the array
elements must be between 0 and 4095, and ISEED(4) must be
odd.
On exit, the seed is updated.
WORK
WORK is REAL array, dimension (M+N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2015
subroutine slagsy (integer N, integer K, real, dimension( * ) D, real, dimension( lda, * ) A,
integer LDA, integer, dimension( 4 ) ISEED, real, dimension( * ) WORK, integer INFO)
SLAGSY
Purpose:
SLAGSY generates a real symmetric matrix A, by pre- and post-
multiplying a real diagonal matrix D with a random orthogonal matrix:
A = U*D*U'. The semi-bandwidth may then be reduced to k by additional
orthogonal transformations.
Parameters:
N
N is INTEGER
The order of the matrix A. N >= 0.
K
K is INTEGER
The number of nonzero subdiagonals within the band of A.
0 <= K <= N-1.
D
D is REAL array, dimension (N)
The diagonal elements of the diagonal matrix D.
A
A is REAL array, dimension (LDA,N)
The generated n by n symmetric matrix A (the full matrix is
stored).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= N.
ISEED
ISEED is INTEGER array, dimension (4)
On entry, the seed of the random number generator; the array
elements must be between 0 and 4095, and ISEED(4) must be
odd.
On exit, the seed is updated.
WORK
WORK is REAL array, dimension (2*N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine slahilb (integer N, integer NRHS, real, dimension(lda, n) A, integer LDA, real,
dimension(ldx, nrhs) X, integer LDX, real, dimension(ldb, nrhs) B, integer LDB, real,
dimension(n) WORK, integer INFO)
SLAHILB
Purpose:
SLAHILB generates an N by N scaled Hilbert matrix in A along with
NRHS right-hand sides in B and solutions in X such that A*X=B.
The Hilbert matrix is scaled by M = LCM(1, 2, ..., 2*N-1) so that all
entries are integers. The right-hand sides are the first NRHS
columns of M * the identity matrix, and the solutions are the
first NRHS columns of the inverse Hilbert matrix.
The condition number of the Hilbert matrix grows exponentially with
its size, roughly as O(e ** (3.5*N)). Additionally, the inverse
Hilbert matrices beyond a relatively small dimension cannot be
generated exactly without extra precision. Precision is exhausted
when the largest entry in the inverse Hilbert matrix is greater than
2 to the power of the number of bits in the fraction of the data type
used plus one, which is 24 for single precision.
In single, the generated solution is exact for N <= 6 and has
small componentwise error for 7 <= N <= 11.
Parameters:
N
N is INTEGER
The dimension of the matrix A.
NRHS
NRHS is INTEGER
The requested number of right-hand sides.
A
A is REAL array, dimension (LDA, N)
The generated scaled Hilbert matrix.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= N.
X
X is REAL array, dimension (LDX, NRHS)
The generated exact solutions. Currently, the first NRHS
columns of the inverse Hilbert matrix.
LDX
LDX is INTEGER
The leading dimension of the array X. LDX >= N.
B
B is REAL array, dimension (LDB, NRHS)
The generated right-hand sides. Currently, the first NRHS
columns of LCM(1, 2, ..., 2*N-1) * the identity matrix.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= N.
WORK
WORK is REAL array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
= 1: N is too large; the data is still generated but may not
be not exact.
< 0: if INFO = -i, the i-th argument had an illegal value
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2015
subroutine slakf2 (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real,
dimension( lda, * ) B, real, dimension( lda, * ) D, real, dimension( lda, * ) E, real,
dimension( ldz, * ) Z, integer LDZ)
SLAKF2
Purpose:
Form the 2*M*N by 2*M*N matrix
Z = [ kron(In, A) -kron(B', Im) ]
[ kron(In, D) -kron(E', Im) ],
where In is the identity matrix of size n and X' is the transpose
of X. kron(X, Y) is the Kronecker product between the matrices X
and Y.
Parameters:
M
M is INTEGER
Size of matrix, must be >= 1.
N
N is INTEGER
Size of matrix, must be >= 1.
A
A is REAL, dimension ( LDA, M )
The matrix A in the output matrix Z.
LDA
LDA is INTEGER
The leading dimension of A, B, D, and E. ( LDA >= M+N )
B
B is REAL, dimension ( LDA, N )
D
D is REAL, dimension ( LDA, M )
E
E is REAL, dimension ( LDA, N )
The matrices used in forming the output matrix Z.
Z
Z is REAL, dimension ( LDZ, 2*M*N )
The resultant Kronecker M*N*2 by M*N*2 matrix (see above.)
LDZ
LDZ is INTEGER
The leading dimension of Z. ( LDZ >= 2*M*N )
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
real function slaran (integer, dimension( 4 ) ISEED)
SLARAN
Purpose:
SLARAN returns a random real number from a uniform (0,1)
distribution.
Parameters:
ISEED
ISEED is INTEGER array, dimension (4)
On entry, the seed of the random number generator; the array
elements must be between 0 and 4095, and ISEED(4) must be
odd.
On exit, the seed is updated.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
Further Details:
This routine uses a multiplicative congruential method with modulus
2**48 and multiplier 33952834046453 (see G.S.Fishman,
'Multiplicative congruential random number generators with modulus
2**b: an exhaustive analysis for b = 32 and a partial analysis for
b = 48', Math. Comp. 189, pp 331-344, 1990).
48-bit integers are stored in 4 integer array elements with 12 bits
per element. Hence the routine is portable across machines with
integers of 32 bits or more.
subroutine slarge (integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( 4
) ISEED, real, dimension( * ) WORK, integer INFO)
SLARGE
Purpose:
SLARGE pre- and post-multiplies a real general n by n matrix A
with a random orthogonal matrix: A = U*D*U'.
Parameters:
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the original n by n matrix A.
On exit, A is overwritten by U*A*U' for some random
orthogonal matrix U.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= N.
ISEED
ISEED is INTEGER array, dimension (4)
On entry, the seed of the random number generator; the array
elements must be between 0 and 4095, and ISEED(4) must be
odd.
On exit, the seed is updated.
WORK
WORK is REAL array, dimension (2*N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
real function slarnd (integer IDIST, integer, dimension( 4 ) ISEED)
SLARND
Purpose:
SLARND returns a random real number from a uniform or normal
distribution.
Parameters:
IDIST
IDIST is INTEGER
Specifies the distribution of the random numbers:
= 1: uniform (0,1)
= 2: uniform (-1,1)
= 3: normal (0,1)
ISEED
ISEED is INTEGER array, dimension (4)
On entry, the seed of the random number generator; the array
elements must be between 0 and 4095, and ISEED(4) must be
odd.
On exit, the seed is updated.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
Further Details:
This routine calls the auxiliary routine SLARAN to generate a random
real number from a uniform (0,1) distribution. The Box-Muller method
is used to transform numbers from a uniform to a normal distribution.
subroutine slaror (character SIDE, character INIT, integer M, integer N, real, dimension( lda,
* ) A, integer LDA, integer, dimension( 4 ) ISEED, real, dimension( * ) X, integer INFO)
SLAROR
Purpose:
SLAROR pre- or post-multiplies an M by N matrix A by a random
orthogonal matrix U, overwriting A. A may optionally be initialized
to the identity matrix before multiplying by U. U is generated using
the method of G.W. Stewart (SIAM J. Numer. Anal. 17, 1980, 403-409).
Parameters:
SIDE
SIDE is CHARACTER*1
Specifies whether A is multiplied on the left or right by U.
= 'L': Multiply A on the left (premultiply) by U
= 'R': Multiply A on the right (postmultiply) by U'
= 'C' or 'T': Multiply A on the left by U and the right
by U' (Here, U' means U-transpose.)
INIT
INIT is CHARACTER*1
Specifies whether or not A should be initialized to the
identity matrix.
= 'I': Initialize A to (a section of) the identity matrix
before applying U.
= 'N': No initialization. Apply U to the input matrix A.
INIT = 'I' may be used to generate square or rectangular
orthogonal matrices:
For M = N and SIDE = 'L' or 'R', the rows will be orthogonal
to each other, as will the columns.
If M < N, SIDE = 'R' produces a dense matrix whose rows are
orthogonal and whose columns are not, while SIDE = 'L'
produces a matrix whose rows are orthogonal, and whose first
M columns are orthogonal, and whose remaining columns are
zero.
If M > N, SIDE = 'L' produces a dense matrix whose columns
are orthogonal and whose rows are not, while SIDE = 'R'
produces a matrix whose columns are orthogonal, and whose
first M rows are orthogonal, and whose remaining rows are
zero.
M
M is INTEGER
The number of rows of A.
N
N is INTEGER
The number of columns of A.
A
A is REAL array, dimension (LDA, N)
On entry, the array A.
On exit, overwritten by U A ( if SIDE = 'L' ),
or by A U ( if SIDE = 'R' ),
or by U A U' ( if SIDE = 'C' or 'T').
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
ISEED
ISEED is INTEGER array, dimension (4)
On entry ISEED specifies the seed of the random number
generator. The array elements should be between 0 and 4095;
if not they will be reduced mod 4096. Also, ISEED(4) must
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 SLAROR to continue the same random number
sequence.
X
X is REAL array, dimension (3*MAX( M, N ))
Workspace of length
2*M + N if SIDE = 'L',
2*N + M if SIDE = 'R',
3*N if SIDE = 'C' or 'T'.
INFO
INFO is INTEGER
An error flag. It is set to:
= 0: normal return
< 0: if INFO = -k, the k-th argument had an illegal value
= 1: if the random numbers generated by SLARND are bad.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine slarot (logical LROWS, logical LLEFT, logical LRIGHT, integer NL, real C, real S,
real, dimension( * ) A, integer LDA, real XLEFT, real XRIGHT)
SLAROT
Purpose:
SLAROT applies a (Givens) rotation to two adjacent rows or
columns, where one element of the first and/or last column/row
for use on matrices stored in some format other than GE, so
that elements of the matrix may be used or modified for which
no array element is provided.
One example is a symmetric matrix in SB format (bandwidth=4), for
which UPLO='L': Two adjacent rows will have the format:
row j: C> C> C> C> C> . . . .
row j+1: C> C> C> C> C> . . . .
'*' indicates elements for which storage is provided,
'.' indicates elements for which no storage is provided, but
are not necessarily zero; their values are determined by
symmetry. ' ' indicates elements which are necessarily zero,
and have no storage provided.
Those columns which have two '*'s can be handled by SROT.
Those columns which have no '*'s can be ignored, since as long
as the Givens rotations are carefully applied to preserve
symmetry, their values are determined.
Those columns which have one '*' have to be handled separately,
by using separate variables "p" and "q":
row j: C> C> C> C> C> p . . .
row j+1: q C> C> C> C> C> . . . .
The element p would have to be set correctly, then that column
is rotated, setting p to its new value. The next call to
SLAROT would rotate columns j and j+1, using p, and restore
symmetry. The element q would start out being zero, and be
made non-zero by the rotation. Later, rotations would presumably
be chosen to zero q out.
Typical Calling Sequences: rotating the i-th and (i+1)-st rows.
------- ------- ---------
General dense matrix:
CALL SLAROT(.TRUE.,.FALSE.,.FALSE., N, C,S,
A(i,1),LDA, DUMMY, DUMMY)
General banded matrix in GB format:
j = MAX(1, i-KL )
NL = MIN( N, i+KU+1 ) + 1-j
CALL SLAROT( .TRUE., i-KL.GE.1, i+KU.LT.N, NL, C,S,
A(KU+i+1-j,j),LDA-1, XLEFT, XRIGHT )
[ note that i+1-j is just MIN(i,KL+1) ]
Symmetric banded matrix in SY format, bandwidth K,
lower triangle only:
j = MAX(1, i-K )
NL = MIN( K+1, i ) + 1
CALL SLAROT( .TRUE., i-K.GE.1, .TRUE., NL, C,S,
A(i,j), LDA, XLEFT, XRIGHT )
Same, but upper triangle only:
NL = MIN( K+1, N-i ) + 1
CALL SLAROT( .TRUE., .TRUE., i+K.LT.N, NL, C,S,
A(i,i), LDA, XLEFT, XRIGHT )
Symmetric banded matrix in SB format, bandwidth K,
lower triangle only:
[ same as for SY, except:]
. . . .
A(i+1-j,j), LDA-1, XLEFT, XRIGHT )
[ note that i+1-j is just MIN(i,K+1) ]
Same, but upper triangle only:
. . .
A(K+1,i), LDA-1, XLEFT, XRIGHT )
Rotating columns is just the transpose of rotating rows, except
for GB and SB: (rotating columns i and i+1)
GB:
j = MAX(1, i-KU )
NL = MIN( N, i+KL+1 ) + 1-j
CALL SLAROT( .TRUE., i-KU.GE.1, i+KL.LT.N, NL, C,S,
A(KU+j+1-i,i),LDA-1, XTOP, XBOTTM )
[note that KU+j+1-i is just MAX(1,KU+2-i)]
SB: (upper triangle)
. . . . . .
A(K+j+1-i,i),LDA-1, XTOP, XBOTTM )
SB: (lower triangle)
. . . . . .
A(1,i),LDA-1, XTOP, XBOTTM )
LROWS - LOGICAL
If .TRUE., then SLAROT will rotate two rows. If .FALSE.,
then it will rotate two columns.
Not modified.
LLEFT - LOGICAL
If .TRUE., then XLEFT will be used instead of the
corresponding element of A for the first element in the
second row (if LROWS=.FALSE.) or column (if LROWS=.TRUE.)
If .FALSE., then the corresponding element of A will be
used.
Not modified.
LRIGHT - LOGICAL
If .TRUE., then XRIGHT will be used instead of the
corresponding element of A for the last element in the
first row (if LROWS=.FALSE.) or column (if LROWS=.TRUE.) If
.FALSE., then the corresponding element of A will be used.
Not modified.
NL - INTEGER
The length of the rows (if LROWS=.TRUE.) or columns (if
LROWS=.FALSE.) to be rotated. If XLEFT and/or XRIGHT are
used, the columns/rows they are in should be included in
NL, e.g., if LLEFT = LRIGHT = .TRUE., then NL must be at
least 2. The number of rows/columns to be rotated
exclusive of those involving XLEFT and/or XRIGHT may
not be negative, i.e., NL minus how many of LLEFT and
LRIGHT are .TRUE. must be at least zero; if not, XERBLA
will be called.
Not modified.
C, S - REAL
Specify the Givens rotation to be applied. If LROWS is
true, then the matrix ( c s )
(-s c ) is applied from the left;
if false, then the transpose thereof is applied from the
right. For a Givens rotation, C**2 + S**2 should be 1,
but this is not checked.
Not modified.
A - REAL array.
The array containing the rows/columns to be rotated. The
first element of A should be the upper left element to
be rotated.
Read and modified.
LDA - INTEGER
The "effective" leading dimension of A. If A contains
a matrix stored in GE or SY format, then this is just
the leading dimension of A as dimensioned in the calling
routine. If A contains a matrix stored in band (GB or SB)
format, then this should be *one less* than the leading
dimension used in the calling routine. Thus, if
A were dimensioned A(LDA,*) in SLAROT, then A(1,j) would
be the j-th element in the first of the two rows
to be rotated, and A(2,j) would be the j-th in the second,
regardless of how the array may be stored in the calling
routine. [A cannot, however, actually be dimensioned thus,
since for band format, the row number may exceed LDA, which
is not legal FORTRAN.]
If LROWS=.TRUE., then LDA must be at least 1, otherwise
it must be at least NL minus the number of .TRUE. values
in XLEFT and XRIGHT.
Not modified.
XLEFT - REAL
If LLEFT is .TRUE., then XLEFT will be used and modified
instead of A(2,1) (if LROWS=.TRUE.) or A(1,2)
(if LROWS=.FALSE.).
Read and modified.
XRIGHT - REAL
If LRIGHT is .TRUE., then XRIGHT will be used and modified
instead of A(1,NL) (if LROWS=.TRUE.) or A(NL,1)
(if LROWS=.FALSE.).
Read and modified.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine slatm1 (integer MODE, real COND, integer IRSIGN, integer IDIST, integer, dimension(
4 ) ISEED, real, dimension( * ) D, integer N, integer INFO)
SLATM1
Purpose:
SLATM1 computes the entries of D(1..N) as specified by
MODE, COND and IRSIGN. IDIST and ISEED determine the generation
of random numbers. SLATM1 is called by SLATMR to generate
random test matrices for LAPACK programs.
Parameters:
MODE
MODE is INTEGER
On entry describes how D is to be computed:
MODE = 0 means do not change D.
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
COND is REAL
On entry, used as described under MODE above.
If used, it must be >= 1. Not modified.
IRSIGN
IRSIGN is INTEGER
On entry, if MODE neither -6, 0 nor 6, determines sign of
entries of D
0 => leave entries of D unchanged
1 => multiply each entry of D by 1 or -1 with probability .5
IDIST
IDIST is INTEGER
On entry, IDIST specifies the type of distribution to be
used to generate a random matrix .
1 => UNIFORM( 0, 1 )
2 => UNIFORM( -1, 1 )
3 => NORMAL( 0, 1 )
Not modified.
ISEED
ISEED is INTEGER array, dimension ( 4 )
On entry ISEED specifies the seed of the random number
generator. 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 SLATM1
to continue the same random number sequence.
Changed on exit.
D
D is REAL array, dimension ( N )
Array to be computed according to MODE, COND and IRSIGN.
May be changed on exit if MODE is nonzero.
N
N is INTEGER
Number of entries of D. Not modified.
INFO
INFO is INTEGER
0 => normal termination
-1 => if MODE not in range -6 to 6
-2 => if MODE neither -6, 0 nor 6, and
IRSIGN neither 0 nor 1
-3 => if MODE neither -6, 0 nor 6 and COND less than 1
-4 => if MODE equals 6 or -6 and IDIST not in range 1 to 3
-7 => if N negative
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2015
real function slatm2 (integer M, integer N, integer I, integer J, integer KL, integer KU,
integer IDIST, integer, dimension( 4 ) ISEED, real, dimension( * ) D, integer IGRADE,
real, dimension( * ) DL, real, dimension( * ) DR, integer IPVTNG, integer, dimension( * )
IWORK, real SPARSE)
SLATM2
Purpose:
SLATM2 returns the (I,J) entry of a random matrix of dimension
(M, N) described by the other parameters. It is called by the
SLATMR routine in order to build random test matrices. No error
checking on parameters is done, because this routine is called in
a tight loop by SLATMR which has already checked the parameters.
Use of SLATM2 differs from SLATM3 in the order in which the random
number generator is called to fill in random matrix entries.
With SLATM2, the generator is called to fill in the pivoted matrix
columnwise. With SLATM3, the generator is called to fill in the
matrix columnwise, after which it is pivoted. Thus, SLATM3 can
be used to construct random matrices which differ only in their
order of rows and/or columns. SLATM2 is used to construct band
matrices while avoiding calling the random number generator for
entries outside the band (and therefore generating random numbers
The matrix whose (I,J) entry is returned is constructed as
follows (this routine only computes one entry):
If I is outside (1..M) or J is outside (1..N), return zero
(this is convenient for generating matrices in band format).
Generate a matrix A with random entries of distribution IDIST.
Set the diagonal to D.
Grade the matrix, if desired, from the left (by DL) and/or
from the right (by DR or DL) as specified by IGRADE.
Permute, if desired, the rows and/or columns as specified by
IPVTNG and IWORK.
Band the matrix to have lower bandwidth KL and upper
bandwidth KU.
Set random entries to zero as specified by SPARSE.
Parameters:
M
M is INTEGER
Number of rows of matrix. Not modified.
N
N is INTEGER
Number of columns of matrix. Not modified.
I
I is INTEGER
Row of entry to be returned. Not modified.
J
J is INTEGER
Column of entry to be returned. Not modified.
KL
KL is INTEGER
Lower bandwidth. Not modified.
KU
KU is INTEGER
Upper bandwidth. Not modified.
IDIST
IDIST is INTEGER
On entry, IDIST specifies the type of distribution to be
used to generate a random matrix .
1 => UNIFORM( 0, 1 )
2 => UNIFORM( -1, 1 )
3 => NORMAL( 0, 1 )
Not modified.
ISEED
ISEED is INTEGER array of dimension ( 4 )
Seed for random number generator.
Changed on exit.
D
D is REAL array of dimension ( MIN( I , J ) )
Diagonal entries of matrix. Not modified.
IGRADE
IGRADE is INTEGER
Specifies grading of matrix as follows:
0 => no grading
1 => matrix premultiplied by diag( DL )
2 => matrix postmultiplied by diag( DR )
3 => matrix premultiplied by diag( DL ) and
postmultiplied by diag( DR )
4 => matrix premultiplied by diag( DL ) and
postmultiplied by inv( diag( DL ) )
5 => matrix premultiplied by diag( DL ) and
postmultiplied by diag( DL )
Not modified.
DL
DL is REAL array ( I or J, as appropriate )
Left scale factors for grading matrix. Not modified.
DR
DR is REAL array ( I or J, as appropriate )
Right scale factors for grading matrix. Not modified.
IPVTNG
IPVTNG is INTEGER
On entry specifies pivoting permutations as follows:
0 => none.
1 => row pivoting.
2 => column pivoting.
3 => full pivoting, i.e., on both sides.
Not modified.
IWORK
IWORK is INTEGER array ( I or J, as appropriate )
This array specifies the permutation used. The
row (or column) in position K was originally in
position IWORK( K ).
This differs from IWORK for SLATM3. Not modified.
SPARSE
SPARSE is REAL between 0. and 1.
On entry specifies the sparsity of the matrix
if sparse matix is to be generated.
SPARSE should lie between 0 and 1.
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.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
real function slatm3 (integer M, integer N, integer I, integer J, integer ISUB, integer JSUB,
integer KL, integer KU, integer IDIST, integer, dimension( 4 ) ISEED, real, dimension( * )
D, integer IGRADE, real, dimension( * ) DL, real, dimension( * ) DR, integer IPVTNG,
integer, dimension( * ) IWORK, real SPARSE)
SLATM3
Purpose:
SLATM3 returns the (ISUB,JSUB) entry of a random matrix of
dimension (M, N) described by the other parameters. (ISUB,JSUB)
is the final position of the (I,J) entry after pivoting
according to IPVTNG and IWORK. SLATM3 is called by the
SLATMR routine in order to build random test matrices. No error
checking on parameters is done, because this routine is called in
a tight loop by SLATMR which has already checked the parameters.
Use of SLATM3 differs from SLATM2 in the order in which the random
number generator is called to fill in random matrix entries.
With SLATM2, the generator is called to fill in the pivoted matrix
columnwise. With SLATM3, the generator is called to fill in the
matrix columnwise, after which it is pivoted. Thus, SLATM3 can
be used to construct random matrices which differ only in their
order of rows and/or columns. SLATM2 is used to construct band
matrices while avoiding calling the random number generator for
entries outside the band (and therefore generating random numbers
in different orders for different pivot orders).
The matrix whose (ISUB,JSUB) entry is returned is constructed as
follows (this routine only computes one entry):
If ISUB is outside (1..M) or JSUB is outside (1..N), return zero
(this is convenient for generating matrices in band format).
Generate a matrix A with random entries of distribution IDIST.
Set the diagonal to D.
Grade the matrix, if desired, from the left (by DL) and/or
from the right (by DR or DL) as specified by IGRADE.
Permute, if desired, the rows and/or columns as specified by
IPVTNG and IWORK.
Band the matrix to have lower bandwidth KL and upper
bandwidth KU.
Set random entries to zero as specified by SPARSE.
Parameters:
M
M is INTEGER
Number of rows of matrix. Not modified.
N
N is INTEGER
Number of columns of matrix. Not modified.
I
I is INTEGER
Row of unpivoted entry to be returned. Not modified.
J
J is INTEGER
Column of unpivoted entry to be returned. Not modified.
ISUB
ISUB is INTEGER
Row of pivoted entry to be returned. Changed on exit.
JSUB
JSUB is INTEGER
Column of pivoted entry to be returned. Changed on exit.
KL
KL is INTEGER
Lower bandwidth. Not modified.
KU
KU is INTEGER
Upper bandwidth. Not modified.
IDIST
IDIST is INTEGER
On entry, IDIST specifies the type of distribution to be
used to generate a random matrix .
1 => UNIFORM( 0, 1 )
2 => UNIFORM( -1, 1 )
3 => NORMAL( 0, 1 )
Not modified.
ISEED
ISEED is INTEGER array of dimension ( 4 )
Seed for random number generator.
Changed on exit.
D
D is REAL array of dimension ( MIN( I , J ) )
Diagonal entries of matrix. Not modified.
IGRADE
IGRADE is INTEGER
Specifies grading of matrix as follows:
0 => no grading
1 => matrix premultiplied by diag( DL )
2 => matrix postmultiplied by diag( DR )
3 => matrix premultiplied by diag( DL ) and
postmultiplied by diag( DR )
4 => matrix premultiplied by diag( DL ) and
postmultiplied by inv( diag( DL ) )
5 => matrix premultiplied by diag( DL ) and
postmultiplied by diag( DL )
Not modified.
DL
DL is REAL array ( I or J, as appropriate )
Left scale factors for grading matrix. Not modified.
DR
DR is REAL array ( I or J, as appropriate )
Right scale factors for grading matrix. Not modified.
IPVTNG
IPVTNG is INTEGER
On entry specifies pivoting permutations as follows:
0 => none.
1 => row pivoting.
2 => column pivoting.
3 => full pivoting, i.e., on both sides.
Not modified.
IWORK
IWORK is INTEGER array ( I or J, as appropriate )
This array specifies the permutation used. The
row (or column) originally in position K is in
position IWORK( K ) after pivoting.
This differs from IWORK for SLATM2. Not modified.
SPARSE
SPARSE is REAL between 0. and 1.
On entry specifies the sparsity of the matrix
if sparse matix is to be generated.
SPARSE should lie between 0 and 1.
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.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine slatm5 (integer PRTYPE, integer M, integer N, real, dimension( lda, * ) A, integer
LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldc, * ) C, integer LDC,
real, dimension( ldd, * ) D, integer LDD, real, dimension( lde, * ) E, integer LDE, real,
dimension( ldf, * ) F, integer LDF, real, dimension( ldr, * ) R, integer LDR, real,
dimension( ldl, * ) L, integer LDL, real ALPHA, integer QBLCKA, integer QBLCKB)
SLATM5
Purpose:
SLATM5 generates matrices involved in the Generalized Sylvester
equation:
A * R - L * B = C
D * R - L * E = F
They also satisfy (the diagonalization condition)
[ I -L ] ( [ A -C ], [ D -F ] ) [ I R ] = ( [ A ], [ D ] )
[ I ] ( [ B ] [ E ] ) [ I ] ( [ B ] [ E ] )
Parameters:
PRTYPE
PRTYPE is INTEGER
"Points" to a certain type of the matrices to generate
(see futher details).
M
M is INTEGER
Specifies the order of A and D and the number of rows in
C, F, R and L.
N
N is INTEGER
Specifies the order of B and E and the number of columns in
C, F, R and L.
A
A is REAL array, dimension (LDA, M).
On exit A M-by-M is initialized according to PRTYPE.
LDA
LDA is INTEGER
The leading dimension of A.
B
B is REAL array, dimension (LDB, N).
On exit B N-by-N is initialized according to PRTYPE.
LDB
LDB is INTEGER
The leading dimension of B.
C
C is REAL array, dimension (LDC, N).
On exit C M-by-N is initialized according to PRTYPE.
LDC
LDC is INTEGER
The leading dimension of C.
D
D is REAL array, dimension (LDD, M).
On exit D M-by-M is initialized according to PRTYPE.
LDD
LDD is INTEGER
The leading dimension of D.
E
E is REAL array, dimension (LDE, N).
On exit E N-by-N is initialized according to PRTYPE.
LDE
LDE is INTEGER
The leading dimension of E.
F
F is REAL array, dimension (LDF, N).
On exit F M-by-N is initialized according to PRTYPE.
LDF
LDF is INTEGER
The leading dimension of F.
R
R is REAL array, dimension (LDR, N).
On exit R M-by-N is initialized according to PRTYPE.
LDR
LDR is INTEGER
The leading dimension of R.
L
L is REAL array, dimension (LDL, N).
On exit L M-by-N is initialized according to PRTYPE.
LDL
LDL is INTEGER
The leading dimension of L.
ALPHA
ALPHA is REAL
Parameter used in generating PRTYPE = 1 and 5 matrices.
QBLCKA
QBLCKA is INTEGER
When PRTYPE = 3, specifies the distance between 2-by-2
blocks on the diagonal in A. Otherwise, QBLCKA is not
referenced. QBLCKA > 1.
QBLCKB
QBLCKB is INTEGER
When PRTYPE = 3, specifies the distance between 2-by-2
blocks on the diagonal in B. Otherwise, QBLCKB is not
referenced. QBLCKB > 1.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
Further Details:
PRTYPE = 1: A and B are Jordan blocks, D and E are identity matrices
A : if (i == j) then A(i, j) = 1.0
if (j == i + 1) then A(i, j) = -1.0
else A(i, j) = 0.0, i, j = 1...M
B : if (i == j) then B(i, j) = 1.0 - ALPHA
if (j == i + 1) then B(i, j) = 1.0
else B(i, j) = 0.0, i, j = 1...N
D : if (i == j) then D(i, j) = 1.0
else D(i, j) = 0.0, i, j = 1...M
E : if (i == j) then E(i, j) = 1.0
else E(i, j) = 0.0, i, j = 1...N
L = R are chosen from [-10...10],
which specifies the right hand sides (C, F).
PRTYPE = 2 or 3: Triangular and/or quasi- triangular.
A : if (i <= j) then A(i, j) = [-1...1]
else A(i, j) = 0.0, i, j = 1...M
if (PRTYPE = 3) then
A(k + 1, k + 1) = A(k, k)
A(k + 1, k) = [-1...1]
sign(A(k, k + 1) = -(sin(A(k + 1, k))
k = 1, M - 1, QBLCKA
B : if (i <= j) then B(i, j) = [-1...1]
else B(i, j) = 0.0, i, j = 1...N
if (PRTYPE = 3) then
B(k + 1, k + 1) = B(k, k)
B(k + 1, k) = [-1...1]
sign(B(k, k + 1) = -(sign(B(k + 1, k))
k = 1, N - 1, QBLCKB
D : if (i <= j) then D(i, j) = [-1...1].
else D(i, j) = 0.0, i, j = 1...M
E : if (i <= j) then D(i, j) = [-1...1]
else E(i, j) = 0.0, i, j = 1...N
L, R are chosen from [-10...10],
which specifies the right hand sides (C, F).
PRTYPE = 4 Full
A(i, j) = [-10...10]
D(i, j) = [-1...1] i,j = 1...M
B(i, j) = [-10...10]
E(i, j) = [-1...1] i,j = 1...N
R(i, j) = [-10...10]
L(i, j) = [-1...1] i = 1..M ,j = 1...N
L, R specifies the right hand sides (C, F).
PRTYPE = 5 special case common and/or close eigs.
subroutine slatm6 (integer TYPE, integer N, real, dimension( lda, * ) A, integer LDA, real,
dimension( lda, * ) B, real, dimension( ldx, * ) X, integer LDX, real, dimension( ldy, * )
Y, integer LDY, real ALPHA, real BETA, real WX, real WY, real, dimension( * ) S, real,
dimension( * ) DIF)
SLATM6
Purpose:
SLATM6 generates test matrices for the generalized eigenvalue
problem, their corresponding right and left eigenvector matrices,
and also reciprocal condition numbers for all eigenvalues and
the reciprocal condition numbers of eigenvectors corresponding to
the 1th and 5th eigenvalues.
Test Matrices
=============
Two kinds of test matrix pairs
(A, B) = inverse(YH) * (Da, Db) * inverse(X)
are used in the tests:
Type 1:
Da = 1+a 0 0 0 0 Db = 1 0 0 0 0
0 2+a 0 0 0 0 1 0 0 0
0 0 3+a 0 0 0 0 1 0 0
0 0 0 4+a 0 0 0 0 1 0
0 0 0 0 5+a , 0 0 0 0 1 , and
Type 2:
Da = 1 -1 0 0 0 Db = 1 0 0 0 0
1 1 0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 0 1+a 1+b 0 0 0 1 0
0 0 0 -1-b 1+a , 0 0 0 0 1 .
In both cases the same inverse(YH) and inverse(X) are used to compute
(A, B), giving the exact eigenvectors to (A,B) as (YH, X):
YH: = 1 0 -y y -y X = 1 0 -x -x x
0 1 -y y -y 0 1 x -x -x
0 0 1 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0 1 0
0 0 0 0 1, 0 0 0 0 1 ,
where a, b, x and y will have all values independently of each other.
Parameters:
TYPE
TYPE is INTEGER
Specifies the problem type (see futher details).
N
N is INTEGER
Size of the matrices A and B.
A
A is REAL array, dimension (LDA, N).
On exit A N-by-N is initialized according to TYPE.
LDA
LDA is INTEGER
The leading dimension of A and of B.
B
B is REAL array, dimension (LDA, N).
On exit B N-by-N is initialized according to TYPE.
X
X is REAL array, dimension (LDX, N).
On exit X is the N-by-N matrix of right eigenvectors.
LDX
LDX is INTEGER
The leading dimension of X.
Y
Y is REAL array, dimension (LDY, N).
On exit Y is the N-by-N matrix of left eigenvectors.
LDY
LDY is INTEGER
The leading dimension of Y.
ALPHA
ALPHA is REAL
BETA
BETA is REAL
Weighting constants for matrix A.
WX
WX is REAL
Constant for right eigenvector matrix.
WY
WY is REAL
Constant for left eigenvector matrix.
S
S is REAL array, dimension (N)
S(i) is the reciprocal condition number for eigenvalue i.
DIF
DIF is REAL array, dimension (N)
DIF(i) is the reciprocal condition number for eigenvector i.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine slatm7 (integer MODE, real COND, integer IRSIGN, integer IDIST, integer, dimension(
4 ) ISEED, real, dimension( * ) D, integer N, integer RANK, integer INFO)
SLATM7
Purpose:
SLATM7 computes the entries of D as specified by MODE
COND and IRSIGN. IDIST and ISEED determine the generation
of random numbers. SLATM7 is called by SLATMT to generate
random test matrices.
MODE - INTEGER
On entry describes how D is to be computed:
MODE = 0 means do not change D.
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)) I=1:RANK
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 - REAL
On entry, used as described under MODE above.
If used, it must be >= 1. Not modified.
IRSIGN - INTEGER
On entry, if MODE neither -6, 0 nor 6, determines sign of
entries of D
0 => leave entries of D unchanged
1 => multiply each entry of D by 1 or -1 with probability .5
IDIST - CHARACTER*1
On entry, IDIST specifies the type of distribution to be
used to generate a random matrix .
1 => UNIFORM( 0, 1 )
2 => UNIFORM( -1, 1 )
3 => NORMAL( 0, 1 )
Not modified.
ISEED - INTEGER array, dimension ( 4 )
On entry ISEED specifies the seed of the random number
generator. 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 SLATM7
to continue the same random number sequence.
Changed on exit.
D - REAL array, dimension ( MIN( M , N ) )
Array to be computed according to MODE, COND and IRSIGN.
May be changed on exit if MODE is nonzero.
N - INTEGER
Number of entries of D. Not modified.
RANK - INTEGER
The rank of matrix to be generated for modes 1,2,3 only.
D( RANK+1:N ) = 0.
Not modified.
INFO - INTEGER
0 => normal termination
-1 => if MODE not in range -6 to 6
-2 => if MODE neither -6, 0 nor 6, and
IRSIGN neither 0 nor 1
-3 => if MODE neither -6, 0 nor 6 and COND less than 1
-4 => if MODE equals 6 or -6 and IDIST not in range 1 to 3
-7 => if N negative
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
September 2012
subroutine slatme (integer N, character DIST, integer, dimension( 4 ) ISEED, real, dimension(
* ) D, integer MODE, real COND, real DMAX, character, dimension( * ) EI, character RSIGN,
character UPPER, character SIM, real, dimension( * ) DS, integer MODES, real CONDS,
integer KL, integer KU, real ANORM, real, dimension( lda, * ) A, integer LDA, real,
dimension( * ) WORK, integer INFO)
SLATME
Purpose:
SLATME generates random non-symmetric square matrices with
specified eigenvalues for testing LAPACK programs.
SLATME operates by applying the following sequence of
operations:
1. Set the diagonal to D, where D may be input or
computed according to MODE, COND, DMAX, and RSIGN
as described below.
2. If complex conjugate pairs are desired (MODE=0 and EI(1)='R',
or MODE=5), certain pairs of adjacent elements of D are
interpreted as the real and complex parts of a complex
conjugate pair; A thus becomes block diagonal, with 1x1
and 2x2 blocks.
3. If UPPER='T', the upper triangle of A is set to random values
out of distribution DIST.
4. If SIM='T', A is multiplied on the left by a random matrix
X, whose singular values are specified by DS, MODES, and
CONDS, and on the right by X inverse.
5. If KL < N-1, the lower bandwidth is reduced to KL using
Householder transformations. If KU < N-1, the upper
bandwidth is reduced to KU.
6. If ANORM is not negative, the matrix is scaled to have
maximum-element-norm ANORM.
(Note: since the matrix cannot be reduced beyond Hessenberg form,
no packing options are available.)
Parameters:
N
N is INTEGER
The number of columns (or rows) of A. Not modified.
DIST
DIST is CHARACTER*1
On entry, DIST specifies the type of distribution to be used
to generate the random eigen-/singular values, and for the
upper triangle (see UPPER).
'U' => UNIFORM( 0, 1 ) ( 'U' for uniform )
'S' => UNIFORM( -1, 1 ) ( 'S' for symmetric )
'N' => NORMAL( 0, 1 ) ( 'N' for normal )
Not modified.
ISEED
ISEED is 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 SLATME
to continue the same random number sequence.
Changed on exit.
D
D is REAL array, dimension ( N )
This array is used to specify the eigenvalues of A. If
MODE=0, then D is assumed to contain the eigenvalues (but
see the description of EI), otherwise they will be
computed according to MODE, COND, DMAX, and RSIGN and
placed in D.
Modified if MODE is nonzero.
MODE
MODE is INTEGER
On entry this describes how the eigenvalues are to
be specified:
MODE = 0 means use D (with EI) 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. Each odd-even pair
of elements will be either used as two real
eigenvalues or as the real and imaginary part
of a complex conjugate pair of eigenvalues;
the choice of which is done is random, with
50-50 probability, for each pair.
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 between 1 and 4, D has entries ranging
from 1 to 1/COND, if between -1 and -4, D has entries
ranging from 1/COND to 1,
Not modified.
COND
COND is REAL
On entry, this is used as described under MODE above.
If used, it must be >= 1. Not modified.
DMAX
DMAX is 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))). 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.
EI
EI is CHARACTER*1 array, dimension ( N )
If MODE is 0, and EI(1) is not ' ' (space character),
this array specifies which elements of D (on input) are
real eigenvalues and which are the real and imaginary parts
of a complex conjugate pair of eigenvalues. The elements
of EI may then only have the values 'R' and 'I'. If
EI(j)='R' and EI(j+1)='I', then the j-th eigenvalue is
CMPLX( D(j) , D(j+1) ), and the (j+1)-th is the complex
conjugate thereof. If EI(j)=EI(j+1)='R', then the j-th
eigenvalue is D(j) (i.e., real). EI(1) may not be 'I',
nor may two adjacent elements of EI both have the value 'I'.
If MODE is not 0, then EI is ignored. If MODE is 0 and
EI(1)=' ', then the eigenvalues will all be real.
Not modified.
RSIGN
RSIGN is CHARACTER*1
If MODE is not 0, 6, or -6, and RSIGN='T', then the
elements of D, as computed according to MODE and COND, will
be multiplied by a random sign (+1 or -1). If RSIGN='F',
they will not be. RSIGN may only have the values 'T' or
'F'.
Not modified.
UPPER
UPPER is CHARACTER*1
If UPPER='T', then the elements of A above the diagonal
(and above the 2x2 diagonal blocks, if A has complex
eigenvalues) will be set to random numbers out of DIST.
If UPPER='F', they will not. UPPER may only have the
values 'T' or 'F'.
Not modified.
SIM
SIM is CHARACTER*1
If SIM='T', then A will be operated on by a "similarity
transform", i.e., multiplied on the left by a matrix X and
on the right by X inverse. X = U S V, where U and V are
random unitary matrices and S is a (diagonal) matrix of
singular values specified by DS, MODES, and CONDS. If
SIM='F', then A will not be transformed.
Not modified.
DS
DS is REAL array, dimension ( N )
This array is used to specify the singular values of X,
in the same way that D specifies the eigenvalues of A.
If MODE=0, the DS contains the singular values, which
may not be zero.
Modified if MODE is nonzero.
MODES
MODES is INTEGER
CONDS
CONDS is REAL
Same as MODE and COND, but for specifying the diagonal
of S. MODES=-6 and +6 are not allowed (since they would
result in randomly ill-conditioned eigenvalues.)
KL
KL is INTEGER
This specifies the lower bandwidth of the matrix. KL=1
specifies upper Hessenberg form. If KL is at least N-1,
then A will have full lower bandwidth. KL must be at
least 1.
Not modified.
KU
KU is INTEGER
This specifies the upper bandwidth of the matrix. KU=1
specifies lower Hessenberg form. If KU is at least N-1,
then A will have full upper bandwidth; if KU and KL
are both at least N-1, then A will be dense. Only one of
KU and KL may be less than N-1. KU must be at least 1.
Not modified.
ANORM
ANORM is REAL
If ANORM is not negative, then A will be scaled by a non-
negative real number to make the maximum-element-norm of A
to be ANORM.
Not modified.
A
A is REAL array, dimension ( LDA, N )
On exit A is the desired test matrix.
Modified.
LDA
LDA is INTEGER
LDA specifies the first dimension of A as declared in the
calling program. LDA must be at least N.
Not modified.
WORK
WORK is REAL array, dimension ( 3*N )
Workspace.
Modified.
INFO
INFO is INTEGER
Error code. On exit, INFO will be set to one of the
following values:
0 => normal return
-1 => N negative
-2 => DIST illegal string
-5 => MODE not in range -6 to 6
-6 => COND less than 1.0, and MODE neither -6, 0 nor 6
-8 => EI(1) is not ' ' or 'R', EI(j) is not 'R' or 'I', or
two adjacent elements of EI are 'I'.
-9 => RSIGN is not 'T' or 'F'
-10 => UPPER is not 'T' or 'F'
-11 => SIM is not 'T' or 'F'
-12 => MODES=0 and DS has a zero singular value.
-13 => MODES is not in the range -5 to 5.
-14 => MODES is nonzero and CONDS is less than 1.
-15 => KL is less than 1.
-16 => KU is less than 1, or KL and KU are both less than
N-1.
-19 => LDA is less than N.
1 => Error return from SLATM1 (computing D)
2 => Cannot scale to DMAX (max. eigenvalue is 0)
3 => Error return from SLATM1 (computing DS)
4 => Error return from SLARGE
5 => Zero singular value from SLATM1.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine slatmr (integer M, integer N, character DIST, integer, dimension( 4 ) ISEED,
character SYM, real, dimension( * ) D, integer MODE, real COND, real DMAX, character
RSIGN, character GRADE, real, dimension( * ) DL, integer MODEL, real CONDL, real,
dimension( * ) DR, integer MODER, real CONDR, character PIVTNG, integer, dimension( * )
IPIVOT, integer KL, integer KU, real SPARSE, real ANORM, character PACK, real, dimension(
lda, * ) A, integer LDA, integer, dimension( * ) IWORK, integer INFO)
SLATMR
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.
Parameters:
M
M is INTEGER
Number of rows of A. Not modified.
N
N is INTEGER
Number of columns of A. Not modified.
DIST
DIST is 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
ISEED is 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
SYM is CHARACTER*1
If SYM='S' or 'H', generated matrix is symmetric.
If SYM='N', generated matrix is nonsymmetric.
Not modified.
D
D is 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
MODE is 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
COND is REAL
On entry, used as described under MODE above.
If used, it must be >= 1. Not modified.
DMAX
DMAX is 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
RSIGN is 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
GRADE is 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
DL is 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
MODEL is 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
CONDL is REAL
When MODEL is not zero, this specifies the condition number
of the computed DL. Not modified.
DR
DR is 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
MODER is 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
CONDR is REAL
When MODER is not zero, this specifies the condition number
of the computed DR. Not modified.
PIVTNG
PIVTNG is 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
IPIVOT is 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
SPARSE is 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
KL is 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
KU is 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
ANORM is 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
PACK is 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
A is 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
LDA is 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
IWORK is INTEGER array, dimension ( N or M)
Workspace. Not referenced if PIVTNG = 'N'. Changed on exit.
INFO
INFO is 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
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine slatms (integer M, integer N, character DIST, integer, dimension( 4 ) ISEED,
character SYM, real, dimension( * ) D, integer MODE, real COND, real DMAX, integer KL,
integer KU, character PACK, real, dimension( lda, * ) A, integer LDA, real, dimension( * )
WORK, integer INFO)
SLATMS
Purpose:
SLATMS generates random matrices with specified singular values
(or symmetric/hermitian with specified eigenvalues)
for testing LAPACK programs.
SLATMS 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 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 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 symmetric)
zero out lower half (if symmetric)
store the upper half columnwise (if symmetric or upper
triangular)
store the lower half columnwise (if symmetric or lower
triangular)
store the lower triangle in banded format (if symmetric
or lower triangular)
store the upper triangle in banded format (if symmetric
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.
Parameters:
M
M is INTEGER
The number of rows of A. Not modified.
N
N is INTEGER
The number of columns of A. Not modified.
DIST
DIST is 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
ISEED is 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 SLATMS
to continue the same random number sequence.
Changed on exit.
SYM
SYM is CHARACTER*1
If SYM='S' or 'H', the generated matrix is symmetric, with
eigenvalues specified by D, COND, MODE, and DMAX; they
may be positive, negative, or zero.
If SYM='P', the generated matrix is symmetric, 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.
Not modified.
D
D is 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
MODE is 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: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,
If SYM='S' or '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
COND is REAL
On entry, this is used as described under MODE above.
If used, it must be >= 1. Not modified.
DMAX
DMAX is 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.
KL
KL is 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.
Not modified.
KU
KU is 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.
Not modified.
PACK
PACK is CHARACTER*1
This 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 the matrix is symmetric or upper triangular)
'R' => store the lower triangle columnwise
(only if the matrix is symmetric or lower triangular)
'B' => store the lower triangle in band storage scheme
(only if matrix symmetric or lower triangular)
'Q' => store the upper triangle in band storage scheme
(only if matrix symmetric 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 or TB - use 'B' or 'Q'
PP, SP or TP - use 'C' or 'R'
If two calls to SLATMS differ only in the PACK parameter,
they will generate mathematically equivalent matrices.
Not modified.
A
A is REAL 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
LDA is 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
WORK is REAL array, dimension ( 3*MAX( N , M ) )
Workspace.
Modified.
INFO
INFO is 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='S' or 'H' and KU 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 SLATM1
2 => Cannot scale to DMAX (max. sing. value is 0)
3 => Error return from SLAGGE or SLAGSY
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine slatmt (integer M, integer N, character DIST, integer, dimension( 4 ) ISEED,
character SYM, real, dimension( * ) D, integer MODE, real COND, real DMAX, integer RANK,
integer KL, integer KU, character PACK, real, dimension( lda, * ) A, integer LDA, real,
dimension( * ) WORK, integer INFO)
SLATMT
Purpose:
SLATMT generates random matrices with specified singular values
(or symmetric/hermitian with specified eigenvalues)
for testing LAPACK programs.
SLATMT 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 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 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 symmetric)
zero out lower half (if symmetric)
store the upper half columnwise (if symmetric or upper
triangular)
store the lower half columnwise (if symmetric or lower
triangular)
store the lower triangle in banded format (if symmetric
or lower triangular)
store the upper triangle in banded format (if symmetric
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.
Parameters:
M
M is INTEGER
The number of rows of A. Not modified.
N
N is INTEGER
The number of columns of A. Not modified.
DIST
DIST is 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
ISEED is 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 SLATMT
to continue the same random number sequence.
Changed on exit.
SYM
SYM is CHARACTER*1
If SYM='S' or 'H', the generated matrix is symmetric, with
eigenvalues specified by D, COND, MODE, and DMAX; they
may be positive, negative, or zero.
If SYM='P', the generated matrix is symmetric, 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.
Not modified.
D
D is 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
MODE is 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='S' or '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
COND is REAL
On entry, this is used as described under MODE above.
If used, it must be >= 1. Not modified.
DMAX
DMAX is 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
RANK is INTEGER
The rank of matrix to be generated for modes 1,2,3 only.
D( RANK+1:N ) = 0.
Not modified.
KL
KL is 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.
Not modified.
KU
KU is 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.
Not modified.
PACK
PACK is CHARACTER*1
This 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 the matrix is symmetric or upper triangular)
'R' => store the lower triangle columnwise
(only if the matrix is symmetric or lower triangular)
'B' => store the lower triangle in band storage scheme
(only if matrix symmetric or lower triangular)
'Q' => store the upper triangle in band storage scheme
(only if matrix symmetric 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 or TB - use 'B' or 'Q'
PP, SP or TP - use 'C' or 'R'
If two calls to SLATMT differ only in the PACK parameter,
they will generate mathematically equivalent matrices.
Not modified.
A
A is REAL 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
LDA is 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
WORK is REAL array, dimension ( 3*MAX( N , M ) )
Workspace.
Modified.
INFO
INFO is 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='S' or 'H' and KU 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 SLAGGE or SLAGSY
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
Author
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