ZLATME
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
June 2010
June 2010
Purpose
ZLATME generates random non-symmetric square matrices with
specified eigenvalues for testing LAPACK programs.
ZLATME 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 UPPER='T', the upper triangle of A is set to random values
out of distribution DIST.
3. 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.
4. 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.
5. 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.)
specified eigenvalues for testing LAPACK programs.
ZLATME 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 UPPER='T', the upper triangle of A is set to random values
out of distribution DIST.
3. 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.
4. 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.
5. 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.)
Arguments
| N |
(input) INTEGER
The number of columns (or rows) of A. Not modified.
|
| DIST |
(input) CHARACTER*1
On entry, DIST specifies the type of distribution to be used
to generate the random eigen-/singular values, and on 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 ) 'D' => uniform on the complex disc |z| < 1. 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 ZLATME to continue the same random number sequence. Changed on exit. |
| D |
(input/output) COMPLEX*16 array, dimension ( N )
This array is used to specify the eigenvalues of A. If
MODE=0, then D is assumed to contain the eigenvalues otherwise they will be computed according to MODE, COND, DMAX, and RSIGN and placed in D. Modified if MODE is nonzero. |
| MODE |
(input) INTEGER
On entry this describes how the 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 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 |
(input) DOUBLE PRECISION
On entry, this is used as described under MODE above.
If used, it must be >= 1. Not modified. |
| DMAX |
(input) COMPLEX*16
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 or real: if DMAX is negative or complex (or zero), D will be scaled by a negative or complex number (or zero). If RSIGN='F' then the largest (absolute) eigenvalue will be equal to DMAX. Not modified. |
| EI |
(input) CHARACTER*1 array, dimension ( N )
(ignored)
Not modified. |
| RSIGN |
(input) 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 complex number from the unit circle |z| = 1. If RSIGN='F', they will not be. RSIGN may only have the values 'T' or 'F'. Not modified. |
| UPPER |
(input) CHARACTER*1
If UPPER='T', then the elements of A above the diagonal
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 |
(input) 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 |
(input/output) DOUBLE PRECISION 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 |
(input) INTEGER
|
| CONDS |
(input) DOUBLE PRECISION
Similar to 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 |
(input) 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. Not modified. |
| KU |
(input) 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. Not modified. |
| ANORM |
(input) DOUBLE PRECISION
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 |
(output) COMPLEX*16 array, dimension ( LDA, N )
On exit A is the desired test matrix.
Modified. |
| LDA |
(input) INTEGER
LDA specifies the first dimension of A as declared in the
calling program. LDA must be at least M. Not modified. |
| WORK |
(workspace) COMPLEX*16 array, dimension ( 3*N )
Workspace.
Modified. |
| INFO |
(output) 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 -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 M. 1 => Error return from ZLATM1 (computing D) 2 => Cannot scale to DMAX (max. eigenvalue is 0) 3 => Error return from DLATM1 (computing DS) 4 => Error return from ZLARGE 5 => Zero singular value from DLATM1. |