LAPACK  3.4.2 LAPACK: Linear Algebra PACKage
Collaboration diagram for complex16:


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## Functions/Subroutines

subroutine zlagge (M, N, KL, KU, D, A, LDA, ISEED, WORK, INFO)
ZLAGGE
subroutine zlaghe (N, K, D, A, LDA, ISEED, WORK, INFO)
ZLAGHE
subroutine zlagsy (N, K, D, A, LDA, ISEED, WORK, INFO)
ZLAGSY
subroutine zlahilb (N, NRHS, A, LDA, X, LDX, B, LDB, WORK, INFO, PATH)
ZLAHILB
subroutine zlakf2 (M, N, A, LDA, B, D, E, Z, LDZ)
ZLAKF2
subroutine zlarge (N, A, LDA, ISEED, WORK, INFO)
ZLARGE
COMPLEX *16 function zlarnd (IDIST, ISEED)
ZLARND
subroutine zlaror (SIDE, INIT, M, N, A, LDA, ISEED, X, INFO)
ZLAROR
subroutine zlarot (LROWS, LLEFT, LRIGHT, NL, C, S, A, LDA, XLEFT, XRIGHT)
ZLAROT
subroutine zlatm1 (MODE, COND, IRSIGN, IDIST, ISEED, D, N, INFO)
ZLATM1
COMPLEX *16 function zlatm2 (M, N, I, J, KL, KU, IDIST, ISEED, D, IGRADE, DL, DR, IPVTNG, IWORK, SPARSE)
ZLATM2
COMPLEX *16 function zlatm3 (M, N, I, J, ISUB, JSUB, KL, KU, IDIST, ISEED, D, IGRADE, DL, DR, IPVTNG, IWORK, SPARSE)
ZLATM3
subroutine zlatm5 (PRTYPE, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, R, LDR, L, LDL, ALPHA, QBLCKA, QBLCKB)
ZLATM5
subroutine zlatm6 (TYPE, N, A, LDA, B, X, LDX, Y, LDY, ALPHA, BETA, WX, WY, S, DIF)
ZLATM6
subroutine zlatme (N, DIST, ISEED, D, MODE, COND, DMAX, RSIGN, UPPER, SIM, DS, MODES, CONDS, KL, KU, ANORM, A, LDA, WORK, INFO)
ZLATME
subroutine zlatmr (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)
ZLATMR
subroutine zlatms (M, N, DIST, ISEED, SYM, D, MODE, COND, DMAX, KL, KU, PACK, A, LDA, WORK, INFO)
ZLATMS
subroutine zlatmt (M, N, DIST, ISEED, SYM, D, MODE, COND, DMAX, RANK, KL, KU, PACK, A, LDA, WORK, INFO)
ZLATMT

## Detailed Description

This is the group of complex16 LAPACK TESTING MATGEN routines.

## Function/Subroutine Documentation

 subroutine zlagge ( integer M, integer N, integer KL, integer KU, double precision, dimension( * ) D, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( 4 ) ISEED, complex*16, dimension( * ) WORK, integer INFO )

ZLAGGE

Purpose:
``` ZLAGGE generates a complex general m by n matrix A, by pre- and post-
multiplying a real diagonal matrix D with random unitary matrices:
A = U*D*V. The lower and upper bandwidths may then be reduced to
kl and ku by additional unitary transformations.```
Parameters:
 [in] M ``` M is INTEGER The number of rows of the matrix A. M >= 0.``` [in] N ``` N is INTEGER The number of columns of the matrix A. N >= 0.``` [in] KL ``` KL is INTEGER The number of nonzero subdiagonals within the band of A. 0 <= KL <= M-1.``` [in] KU ``` KU is INTEGER The number of nonzero superdiagonals within the band of A. 0 <= KU <= N-1.``` [in] D ``` D is DOUBLE PRECISION array, dimension (min(M,N)) The diagonal elements of the diagonal matrix D.``` [out] A ``` A is COMPLEX*16 array, dimension (LDA,N) The generated m by n matrix A.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= M.``` [in,out] 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.``` [out] WORK ` WORK is COMPLEX*16 array, dimension (M+N)` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Date:
November 2011

Definition at line 115 of file zlagge.f.

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 subroutine zlaghe ( integer N, integer K, double precision, dimension( * ) D, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( 4 ) ISEED, complex*16, dimension( * ) WORK, integer INFO )

ZLAGHE

Purpose:
``` ZLAGHE generates a complex hermitian matrix A, by pre- and post-
multiplying a real diagonal matrix D with a random unitary matrix:
A = U*D*U'. The semi-bandwidth may then be reduced to k by additional
unitary transformations.```
Parameters:
 [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in] K ``` K is INTEGER The number of nonzero subdiagonals within the band of A. 0 <= K <= N-1.``` [in] D ``` D is DOUBLE PRECISION array, dimension (N) The diagonal elements of the diagonal matrix D.``` [out] A ``` A is COMPLEX*16 array, dimension (LDA,N) The generated n by n hermitian matrix A (the full matrix is stored).``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= N.``` [in,out] 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.``` [out] WORK ` WORK is COMPLEX*16 array, dimension (2*N)` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Date:
November 2011

Definition at line 103 of file zlaghe.f.

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 subroutine zlagsy ( integer N, integer K, double precision, dimension( * ) D, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( 4 ) ISEED, complex*16, dimension( * ) WORK, integer INFO )

ZLAGSY

Purpose:
``` ZLAGSY generates a complex symmetric matrix A, by pre- and post-
multiplying a real diagonal matrix D with a random unitary matrix:
A = U*D*U**T. The semi-bandwidth may then be reduced to k by
Parameters:
 [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in] K ``` K is INTEGER The number of nonzero subdiagonals within the band of A. 0 <= K <= N-1.``` [in] D ``` D is DOUBLE PRECISION array, dimension (N) The diagonal elements of the diagonal matrix D.``` [out] A ``` A is COMPLEX*16 array, dimension (LDA,N) The generated n by n symmetric matrix A (the full matrix is stored).``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= N.``` [in,out] 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.``` [out] WORK ` WORK is COMPLEX*16 array, dimension (2*N)` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Date:
November 2011

Definition at line 103 of file zlagsy.f.

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 subroutine zlahilb ( integer N, integer NRHS, complex*16, dimension(lda,n) A, integer LDA, complex*16, dimension(ldx, nrhs) X, integer LDX, complex*16, dimension(ldb, nrhs) B, integer LDB, double precision, dimension(n) WORK, integer INFO, character*3 PATH )

ZLAHILB

Purpose:
``` ZLAHILB 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:
 [in] N ``` N is INTEGER The dimension of the matrix A.``` [in] NRHS ``` NRHS is INTEGER The requested number of right-hand sides.``` [out] A ``` A is COMPLEX array, dimension (LDA, N) The generated scaled Hilbert matrix.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= N.``` [out] X ``` X is COMPLEX array, dimension (LDX, NRHS) The generated exact solutions. Currently, the first NRHS columns of the inverse Hilbert matrix.``` [in] LDX ``` LDX is INTEGER The leading dimension of the array X. LDX >= N.``` [out] 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.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= N.``` [out] WORK ` WORK is REAL array, dimension (N)` [out] 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``` [in] PATH ``` PATH is CHARACTER*3 The LAPACK path name.```
Date:
November 2011

Definition at line 134 of file zlahilb.f.

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 subroutine zlakf2 ( integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( lda, * ) B, complex*16, dimension( lda, * ) D, complex*16, dimension( lda, * ) E, complex*16, dimension( ldz, * ) Z, integer LDZ )

ZLAKF2

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:
 [in] M ``` M is INTEGER Size of matrix, must be >= 1.``` [in] N ``` N is INTEGER Size of matrix, must be >= 1.``` [in] A ``` A is COMPLEX*16, dimension ( LDA, M ) The matrix A in the output matrix Z.``` [in] LDA ``` LDA is INTEGER The leading dimension of A, B, D, and E. ( LDA >= M+N )``` [in] B ` B is COMPLEX*16, dimension ( LDA, N )` [in] D ` D is COMPLEX*16, dimension ( LDA, M )` [in] E ``` E is COMPLEX*16, dimension ( LDA, N ) The matrices used in forming the output matrix Z.``` [out] Z ``` Z is COMPLEX*16, dimension ( LDZ, 2*M*N ) The resultant Kronecker M*N*2 by M*N*2 matrix (see above.)``` [in] LDZ ``` LDZ is INTEGER The leading dimension of Z. ( LDZ >= 2*M*N )```
Date:
November 2011

Definition at line 106 of file zlakf2.f.

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 subroutine zlarge ( integer N, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( 4 ) ISEED, complex*16, dimension( * ) WORK, integer INFO )

ZLARGE

Purpose:
``` ZLARGE pre- and post-multiplies a complex general n by n matrix A
with a random unitary matrix: A = U*D*U'.```
Parameters:
 [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in,out] A ``` A is COMPLEX*16 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 unitary matrix U.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= N.``` [in,out] 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.``` [out] WORK ` WORK is COMPLEX*16 array, dimension (2*N)` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Date:
November 2011

Definition at line 88 of file zlarge.f.

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 COMPLEX*16 function zlarnd ( integer IDIST, integer, dimension( 4 ) ISEED )

ZLARND

Purpose:
``` ZLARND returns a random complex number from a uniform or normal
distribution.```
Parameters:
 [in] IDIST ``` IDIST is INTEGER Specifies the distribution of the random numbers: = 1: real and imaginary parts each uniform (0,1) = 2: real and imaginary parts each uniform (-1,1) = 3: real and imaginary parts each normal (0,1) = 4: uniformly distributed on the disc abs(z) <= 1 = 5: uniformly distributed on the circle abs(z) = 1``` [in,out] 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.```
Date:
November 2011
Further Details:
```  This routine calls the auxiliary routine DLARAN 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.```

Definition at line 76 of file zlarnd.f.

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 subroutine zlaror ( character SIDE, character INIT, integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( 4 ) ISEED, complex*16, dimension( * ) X, integer INFO )

ZLAROR

Purpose:
```    ZLAROR pre- or post-multiplies an M by N matrix A by a random
unitary 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, pp. 403-409 ).
(BLAS-2 version)```
Parameters:
 [in] SIDE ``` SIDE is CHARACTER*1 SIDE specifies whether A is multiplied on the left or right by U. SIDE = 'L' Multiply A on the left (premultiply) by U SIDE = 'R' Multiply A on the right (postmultiply) by UC> SIDE = 'C' Multiply A on the left by U and the right by UC> SIDE = 'T' Multiply A on the left by U and the right by U' Not modified.``` [in] INIT ``` INIT is CHARACTER*1 INIT specifies whether or not A should be initialized to the identity matrix. INIT = 'I' Initialize A to (a section of) the identity matrix before applying U. INIT = 'N' No initialization. Apply U to the input matrix A. INIT = 'I' may be used to generate square (i.e., unitary) or rectangular orthogonal matrices (orthogonality being in the sense of ZDOTC): For square matrices, M=N, and SIDE many be either 'L' or 'R'; the rows will be orthogonal to each other, as will the columns. For rectangular matrices where M < N, SIDE = 'R' will produce a dense matrix whose rows will be orthogonal and whose columns will not, while SIDE = 'L' will produce a matrix whose rows will be orthogonal, and whose first M columns will be orthogonal, the remaining columns being zero. For matrices where M > N, just use the previous explanation, interchanging 'L' and 'R' and "rows" and "columns". Not modified.``` [in] M ``` M is INTEGER Number of rows of A. Not modified.``` [in] N ``` N is INTEGER Number of columns of A. Not modified.``` [in,out] A ``` A is COMPLEX*16 array, dimension ( LDA, N ) Input and output array. Overwritten by U A ( if SIDE = 'L' ) or by A U ( if SIDE = 'R' ) or by U A U* ( if SIDE = 'C') or by U A U' ( if SIDE = 'T') on exit.``` [in] LDA ``` LDA is INTEGER Leading dimension of A. Must be at least MAX ( 1, M ). Not modified.``` [in,out] 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 ZLAROR to continue the same random number sequence. Modified.``` [out] X ``` X is COMPLEX*16 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'. Modified.``` [out] INFO ``` INFO is INTEGER An error flag. It is set to: 0 if no error. 1 if ZLARND returned a bad random number (installation problem) -1 if SIDE is not L, R, C, or T. -3 if M is negative. -4 if N is negative or if SIDE is C or T and N is not equal to M. -6 if LDA is less than M.```
Date:
November 2011

Definition at line 159 of file zlaror.f.

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 subroutine zlarot ( logical LROWS, logical LLEFT, logical LRIGHT, integer NL, complex*16 C, complex*16 S, complex*16, dimension( * ) A, integer LDA, complex*16 XLEFT, complex*16 XRIGHT )

ZLAROT

Purpose:
```    ZLAROT 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 DROT.
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
ZLAROT 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 ZLAROT(.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 ZLAROT( .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 ZLAROT( .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 ZLAROT( .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 ZLAROT( .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 ZLAROT 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   - COMPLEX*16
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 (not conjugated) thereof is
applied from the right.  Note that in contrast to the
output of ZROTG or to most versions of ZROT, both C and S
are complex.  For a Givens rotation, |C|**2 + |S|**2 should
be 1, but this is not checked.
Not modified.

A      - COMPLEX*16 array.
The array containing the rows/columns to be rotated.  The
first element of A should be the upper left element to
be rotated.

LDA    - INTEGER
The "effective" leading dimension of A.  If A contains
a matrix stored in GE, HE, 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, HB, 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 ZLAROT, 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  - COMPLEX*16
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.).

XRIGHT - COMPLEX*16
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.).
Date:
November 2011

Definition at line 229 of file zlarot.f.

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 subroutine zlatm1 ( integer MODE, double precision COND, integer IRSIGN, integer IDIST, integer, dimension( 4 ) ISEED, complex*16, dimension( * ) D, integer N, integer INFO )

ZLATM1

Purpose:
```    ZLATM1 computes the entries of D(1..N) as specified by
MODE, COND and IRSIGN. IDIST and ISEED determine the generation
of random numbers. ZLATM1 is called by CLATMR to generate
random test matrices for LAPACK programs.```
Parameters:
 [in] 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.``` [in] COND ``` COND is DOUBLE PRECISION On entry, used as described under MODE above. If used, it must be >= 1. Not modified.``` [in] 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 random complex number uniformly distributed with absolute value 1``` [in] IDIST ``` IDIST is CHARACTER*1 On entry, IDIST specifies the type of distribution to be used to generate a random matrix . 1 => real and imaginary parts each UNIFORM( 0, 1 ) 2 => real and imaginary parts each UNIFORM( -1, 1 ) 3 => real and imaginary parts each NORMAL( 0, 1 ) 4 => complex number uniform in DISK( 0, 1 ) Not modified.``` [in,out] 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 ZLATM1 to continue the same random number sequence. Changed on exit.``` [in,out] D ``` D is COMPLEX*16 array, dimension ( MIN( M , N ) ) Array to be computed according to MODE, COND and IRSIGN. May be changed on exit if MODE is nonzero.``` [in] N ``` N is INTEGER Number of entries of D. Not modified.``` [out] 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 4 -7 => if N negative```
Date:
November 2011

Definition at line 138 of file zlatm1.f.

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 COMPLEX*16 function zlatm2 ( integer M, integer N, integer I, integer J, integer KL, integer KU, integer IDIST, integer, dimension( 4 ) ISEED, complex*16, dimension( * ) D, integer IGRADE, complex*16, dimension( * ) DL, complex*16, dimension( * ) DR, integer IPVTNG, integer, dimension( * ) IWORK, double precision SPARSE )

ZLATM2

Purpose:
```    ZLATM2 returns the (I,J) entry of a random matrix of dimension
(M, N) described by the other paramters. It is called by the
ZLATMR 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 ZLATMR which has already checked the parameters.

Use of ZLATM2 differs from CLATM3 in the order in which the random
number generator is called to fill in random matrix entries.
With ZLATM2, the generator is called to fill in the pivoted matrix
columnwise. With ZLATM3, the generator is called to fill in the
matrix columnwise, after which it is pivoted. Thus, ZLATM3 can
be used to construct random matrices which differ only in their
order of rows and/or columns. ZLATM2 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:
 [in] M ``` M is INTEGER Number of rows of matrix. Not modified.``` [in] N ``` N is INTEGER Number of columns of matrix. Not modified.``` [in] I ``` I is INTEGER Row of entry to be returned. Not modified.``` [in] J ``` J is INTEGER Column of entry to be returned. Not modified.``` [in] KL ``` KL is INTEGER Lower bandwidth. Not modified.``` [in] KU ``` KU is INTEGER Upper bandwidth. Not modified.``` [in] IDIST ``` IDIST is INTEGER On entry, IDIST specifies the type of distribution to be used to generate a random matrix . 1 => real and imaginary parts each UNIFORM( 0, 1 ) 2 => real and imaginary parts each UNIFORM( -1, 1 ) 3 => real and imaginary parts each NORMAL( 0, 1 ) 4 => complex number uniform in DISK( 0 , 1 ) Not modified.``` [in,out] ISEED ``` ISEED is INTEGER array of dimension ( 4 ) Seed for random number generator. Changed on exit.``` [in] D ``` D is COMPLEX*16 array of dimension ( MIN( I , J ) ) Diagonal entries of matrix. Not modified.``` [in] 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( CONJG(DL) ) 6 => matrix premultiplied by diag( DL ) and postmultiplied by diag( DL ) Not modified.``` [in] DL ``` DL is COMPLEX*16 array ( I or J, as appropriate ) Left scale factors for grading matrix. Not modified.``` [in] DR ``` DR is COMPLEX*16 array ( I or J, as appropriate ) Right scale factors for grading matrix. Not modified.``` [in] 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.``` [out] 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 ZLATM3. Not modified.``` [in] SPARSE ``` SPARSE is DOUBLE PRECISION 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.```
Date:
November 2011

Definition at line 211 of file zlatm2.f.

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 COMPLEX*16 function zlatm3 ( integer M, integer N, integer I, integer J, integer ISUB, integer JSUB, integer KL, integer KU, integer IDIST, integer, dimension( 4 ) ISEED, complex*16, dimension( * ) D, integer IGRADE, complex*16, dimension( * ) DL, complex*16, dimension( * ) DR, integer IPVTNG, integer, dimension( * ) IWORK, double precision SPARSE )

ZLATM3

Purpose:
```    ZLATM3 returns the (ISUB,JSUB) entry of a random matrix of
dimension (M, N) described by the other paramters. (ISUB,JSUB)
is the final position of the (I,J) entry after pivoting
according to IPVTNG and IWORK. ZLATM3 is called by the
ZLATMR 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 ZLATMR which has already checked the parameters.

Use of ZLATM3 differs from CLATM2 in the order in which the random
number generator is called to fill in random matrix entries.
With ZLATM2, the generator is called to fill in the pivoted matrix
columnwise. With ZLATM3, the generator is called to fill in the
matrix columnwise, after which it is pivoted. Thus, ZLATM3 can
be used to construct random matrices which differ only in their
order of rows and/or columns. ZLATM2 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:
 [in] M ``` M is INTEGER Number of rows of matrix. Not modified.``` [in] N ``` N is INTEGER Number of columns of matrix. Not modified.``` [in] I ``` I is INTEGER Row of unpivoted entry to be returned. Not modified.``` [in] J ``` J is INTEGER Column of unpivoted entry to be returned. Not modified.``` [in,out] ISUB ``` ISUB is INTEGER Row of pivoted entry to be returned. Changed on exit.``` [in,out] JSUB ``` JSUB is INTEGER Column of pivoted entry to be returned. Changed on exit.``` [in] KL ``` KL is INTEGER Lower bandwidth. Not modified.``` [in] KU ``` KU is INTEGER Upper bandwidth. Not modified.``` [in] IDIST ``` IDIST is INTEGER On entry, IDIST specifies the type of distribution to be used to generate a random matrix . 1 => real and imaginary parts each UNIFORM( 0, 1 ) 2 => real and imaginary parts each UNIFORM( -1, 1 ) 3 => real and imaginary parts each NORMAL( 0, 1 ) 4 => complex number uniform in DISK( 0 , 1 ) Not modified.``` [in,out] ISEED ``` ISEED is INTEGER array of dimension ( 4 ) Seed for random number generator. Changed on exit.``` [in] D ``` D is COMPLEX*16 array of dimension ( MIN( I , J ) ) Diagonal entries of matrix. Not modified.``` [in] 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( CONJG(DL) ) 6 => matrix premultiplied by diag( DL ) and postmultiplied by diag( DL ) Not modified.``` [in] DL ``` DL is COMPLEX*16 array ( I or J, as appropriate ) Left scale factors for grading matrix. Not modified.``` [in] DR ``` DR is COMPLEX*16 array ( I or J, as appropriate ) Right scale factors for grading matrix. Not modified.``` [in] 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.``` [in] 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 ZLATM2. Not modified.``` [in] SPARSE ``` SPARSE is DOUBLE PRECISION 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.```
Date:
November 2011

Definition at line 228 of file zlatm3.f.

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 subroutine zlatm5 ( integer PRTYPE, integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( ldd, * ) D, integer LDD, complex*16, dimension( lde, * ) E, integer LDE, complex*16, dimension( ldf, * ) F, integer LDF, complex*16, dimension( ldr, * ) R, integer LDR, complex*16, dimension( ldl, * ) L, integer LDL, double precision ALPHA, integer QBLCKA, integer QBLCKB )

ZLATM5

Purpose:
``` ZLATM5 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:
 [in] PRTYPE ``` PRTYPE is INTEGER "Points" to a certian type of the matrices to generate (see futher details).``` [in] M ``` M is INTEGER Specifies the order of A and D and the number of rows in C, F, R and L.``` [in] N ``` N is INTEGER Specifies the order of B and E and the number of columns in C, F, R and L.``` [out] A ``` A is COMPLEX*16 array, dimension (LDA, M). On exit A M-by-M is initialized according to PRTYPE.``` [in] LDA ``` LDA is INTEGER The leading dimension of A.``` [out] B ``` B is COMPLEX*16 array, dimension (LDB, N). On exit B N-by-N is initialized according to PRTYPE.``` [in] LDB ``` LDB is INTEGER The leading dimension of B.``` [out] C ``` C is COMPLEX*16 array, dimension (LDC, N). On exit C M-by-N is initialized according to PRTYPE.``` [in] LDC ``` LDC is INTEGER The leading dimension of C.``` [out] D ``` D is COMPLEX*16 array, dimension (LDD, M). On exit D M-by-M is initialized according to PRTYPE.``` [in] LDD ``` LDD is INTEGER The leading dimension of D.``` [out] E ``` E is COMPLEX*16 array, dimension (LDE, N). On exit E N-by-N is initialized according to PRTYPE.``` [in] LDE ``` LDE is INTEGER The leading dimension of E.``` [out] F ``` F is COMPLEX*16 array, dimension (LDF, N). On exit F M-by-N is initialized according to PRTYPE.``` [in] LDF ``` LDF is INTEGER The leading dimension of F.``` [out] R ``` R is COMPLEX*16 array, dimension (LDR, N). On exit R M-by-N is initialized according to PRTYPE.``` [in] LDR ``` LDR is INTEGER The leading dimension of R.``` [out] L ``` L is COMPLEX*16 array, dimension (LDL, N). On exit L M-by-N is initialized according to PRTYPE.``` [in] LDL ``` LDL is INTEGER The leading dimension of L.``` [in] ALPHA ``` ALPHA is DOUBLE PRECISION Parameter used in generating PRTYPE = 1 and 5 matrices.``` [in] 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.``` [in] 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.```
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.```

Definition at line 267 of file zlatm5.f.

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 subroutine zlatm6 ( integer TYPE, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( lda, * ) B, complex*16, dimension( ldx, * ) X, integer LDX, complex*16, dimension( ldy, * ) Y, integer LDY, complex*16 ALPHA, complex*16 BETA, complex*16 WX, complex*16 WY, double precision, dimension( * ) S, double precision, dimension( * ) DIF )

ZLATM6

Purpose:
``` ZLATM6 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+i   0    0       0       0    Db = 1   0   0   0   0
0   1-i   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)i  0         0   0   0   1   0
0    0    0       0 (1+a)-(1+b)i,   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:
 [in] TYPE ``` TYPE is INTEGER Specifies the problem type (see futher details).``` [in] N ``` N is INTEGER Size of the matrices A and B.``` [out] A ``` A is COMPLEX*16 array, dimension (LDA, N). On exit A N-by-N is initialized according to TYPE.``` [in] LDA ``` LDA is INTEGER The leading dimension of A and of B.``` [out] B ``` B is COMPLEX*16 array, dimension (LDA, N). On exit B N-by-N is initialized according to TYPE.``` [out] X ``` X is COMPLEX*16 array, dimension (LDX, N). On exit X is the N-by-N matrix of right eigenvectors.``` [in] LDX ``` LDX is INTEGER The leading dimension of X.``` [out] Y ``` Y is COMPLEX*16 array, dimension (LDY, N). On exit Y is the N-by-N matrix of left eigenvectors.``` [in] LDY ``` LDY is INTEGER The leading dimension of Y.``` [in] ALPHA ` ALPHA is COMPLEX*16` [in] BETA ``` BETA is COMPLEX*16 \verbatim Weighting constants for matrix A.``` [in] WX ``` WX is COMPLEX*16 Constant for right eigenvector matrix.``` [in] WY ``` WY is COMPLEX*16 Constant for left eigenvector matrix.``` [out] S ``` S is DOUBLE PRECISION array, dimension (N) S(i) is the reciprocal condition number for eigenvalue i.``` [out] DIF ``` DIF is DOUBLE PRECISION array, dimension (N) DIF(i) is the reciprocal condition number for eigenvector i.```
Date:
November 2011

Definition at line 174 of file zlatm6.f.

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 subroutine zlatme ( integer N, character DIST, integer, dimension( 4 ) ISEED, complex*16, dimension( * ) D, integer MODE, double precision COND, complex*16 DMAX, character RSIGN, character UPPER, character SIM, double precision, dimension( * ) DS, integer MODES, double precision CONDS, integer KL, integer KU, double precision ANORM, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) WORK, integer INFO )

ZLATME

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.)```
Parameters:
 [in] N ``` N is INTEGER The number of columns (or rows) of A. Not modified.``` [in] DIST ``` DIST is 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.``` [in,out] 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 ZLATME to continue the same random number sequence. Changed on exit.``` [in,out] D ``` D is 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.``` [in] MODE ``` MODE is 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.``` [in] COND ``` COND is DOUBLE PRECISION On entry, this is used as described under MODE above. If used, it must be >= 1. Not modified.``` [in] DMAX ``` DMAX is 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.``` [in] 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 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.``` [in] UPPER ``` UPPER is 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.``` [in] 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.``` [in,out] DS ``` DS is 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.``` [in] MODES ` MODES is INTEGER` [in] CONDS ``` CONDS is 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.)``` [in] 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. Not modified.``` [in] 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. Not modified.``` [in] ANORM ``` ANORM is 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.``` [out] A ``` A is COMPLEX*16 array, dimension ( LDA, N ) On exit A is the desired test matrix. Modified.``` [in] LDA ``` LDA is INTEGER LDA specifies the first dimension of A as declared in the calling program. LDA must be at least M. Not modified.``` [out] WORK ``` WORK is COMPLEX*16 array, dimension ( 3*N ) Workspace. Modified.``` [out] 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 -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.```
Date:
November 2011

Definition at line 298 of file zlatme.f.

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 subroutine zlatmr ( integer M, integer N, character DIST, integer, dimension( 4 ) ISEED, character SYM, complex*16, dimension( * ) D, integer MODE, double precision COND, complex*16 DMAX, character RSIGN, character GRADE, complex*16, dimension( * ) DL, integer MODEL, double precision CONDL, complex*16, dimension( * ) DR, integer MODER, double precision CONDR, character PIVTNG, integer, dimension( * ) IPIVOT, integer KL, integer KU, double precision SPARSE, double precision ANORM, character PACK, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IWORK, integer INFO )

ZLATMR

Purpose:
```    ZLATMR generates random matrices of various types for testing
LAPACK programs.

ZLATMR operates by applying the following sequence of
operations:

Generate a matrix A with random entries of distribution DIST
which is symmetric if SYM='S', Hermitian if SYM='H', 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 or Hermitian)
zero out lower half (if symmetric or Hermitian)
store the upper half columnwise (if symmetric or Hermitian
or square upper triangular)
store the lower half columnwise (if symmetric or Hermitian
or square lower triangular)
same as upper half rowwise if symmetric
same as conjugate upper half rowwise if Hermitian
store the lower triangle in banded format
(if symmetric or Hermitian)
store the upper triangle in banded format
(if symmetric or Hermitian)
store the entire matrix in banded format

Note: If two calls to ZLATMR differ only in the PACK parameter,
they will generate mathematically equivalent matrices.

If two calls to ZLATMR 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:
 [in] M ``` M is INTEGER Number of rows of A. Not modified.``` [in] N ``` N is INTEGER Number of columns of A. Not modified.``` [in] DIST ``` DIST is CHARACTER*1 On entry, DIST specifies the type of distribution to be used to generate a random matrix . 'U' => real and imaginary parts are independent UNIFORM( 0, 1 ) ( 'U' for uniform ) 'S' => real and imaginary parts are independent UNIFORM( -1, 1 ) ( 'S' for symmetric ) 'N' => real and imaginary parts are independent NORMAL( 0, 1 ) ( 'N' for normal ) 'D' => uniform on interior of unit disk ( 'D' for disk ) Not modified.``` [in,out] 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 ZLATMR to continue the same random number sequence. Changed on exit.``` [in] SYM ``` SYM is CHARACTER*1 If SYM='S', generated matrix is symmetric. If SYM='H', generated matrix is Hermitian. If SYM='N', generated matrix is nonsymmetric. Not modified.``` [in,out] D ``` D is COMPLEX*16 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. If the matrix is Hermitian, the real part of D will be taken. May be changed on exit if MODE is nonzero.``` [in] 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.``` [in] COND ``` COND is DOUBLE PRECISION On entry, used as described under MODE above. If used, it must be >= 1. Not modified.``` [in] DMAX ``` DMAX is COMPLEX*16 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 complex (or zero), diagonal will be scaled by a complex number (or zero).``` [in] RSIGN ``` RSIGN is CHARACTER*1 If MODE neither -6, 0 nor 6, specifies sign of diagonal as follows: 'T' => diagonal entries are multiplied by a random complex number uniformly distributed with absolute value 1 'F' => diagonal unchanged Not modified.``` [in] 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) 'H' => matrix premultiplied by diag( DL ) and postmultiplied by diag( CONJG(DL) ) (only if matrix Hermitian or nonsymmetric) 'S' => matrix premultiplied by diag( DL ) and postmultiplied by diag( DL ) (only if matrix symmetric or nonsymmetric) 'E' => matrix premultiplied by diag( DL ) and postmultiplied by inv( diag( DL ) ) ( 'S' for similarity ) (only if matrix nonsymmetric) Note: if GRADE='S', then M must equal N. Not modified.``` [in,out] DL ``` DL is COMPLEX*16 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.``` [in] 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.``` [in] CONDL ``` CONDL is DOUBLE PRECISION When MODEL is not zero, this specifies the condition number of the computed DL. Not modified.``` [in,out] DR ``` DR is COMPLEX*16 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' or 'S'. Changed on exit.``` [in] 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.``` [in] CONDR ``` CONDR is DOUBLE PRECISION When MODER is not zero, this specifies the condition number of the computed DR. Not modified.``` [in] 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 ZLATMR 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.``` [in] 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, ZLATMR 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.``` [in] SPARSE ``` SPARSE is DOUBLE PRECISION 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.``` [in] 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 or Hermitian. Not modified.``` [in] 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 or Hermitian. Not modified.``` [in] ANORM ``` ANORM is DOUBLE PRECISION 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.``` [in] PACK ``` PACK is CHARACTER*1 On entry specifies packing of matrix as follows: 'N' => no packing 'U' => zero out all subdiagonal entries (if symmetric or Hermitian) 'L' => zero out all superdiagonal entries (if symmetric or Hermitian) 'C' => store the upper triangle columnwise (only if matrix symmetric or Hermitian or square upper triangular) 'R' => store the lower triangle columnwise (only if matrix symmetric or Hermitian or square lower triangular) (same as upper half rowwise if symmetric) (same as conjugate upper half rowwise if Hermitian) 'B' => store the lower triangle in band storage scheme (only if matrix symmetric or Hermitian) 'Q' => store the upper triangle in band storage scheme (only if matrix symmetric or Hermitian) '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, HB or TB - use 'B' or 'Q' PP, HP or TP - use 'C' or 'R' If two calls to ZLATMR differ only in the PACK parameter, they will generate mathematically equivalent matrices. Not modified.``` [in,out] A ``` A is COMPLEX*16 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.``` [in] 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.``` [out] IWORK ``` IWORK is INTEGER array, dimension (N or M) Workspace. Not referenced if PIVTNG = 'N'. Changed on exit.``` [out] 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', 'S' or 'E' and SYM = 'H', or GRADE='L', 'R', 'B', 'H' or 'E' and SYM = 'S' -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 ZLATM1 (computing D) 2 => Cannot scale diagonal to DMAX (max. entry is 0) 3 => Error return from ZLATM1 (computing DL) 4 => Error return from ZLATM1 (computing DR) 5 => ANORM is positive, but matrix constructed prior to attempting to scale it to have norm ANORM, is zero```
Date:
November 2011

Definition at line 488 of file zlatmr.f.

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 subroutine zlatms ( integer M, integer N, character DIST, integer, dimension( 4 ) ISEED, character SYM, double precision, dimension( * ) D, integer MODE, double precision COND, double precision DMAX, integer KL, integer KU, character PACK, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) WORK, integer INFO )

ZLATMS

Purpose:
```    ZLATMS generates random matrices with specified singular values
(or hermitian with specified eigenvalues)
for testing LAPACK programs.

ZLATMS operates by applying the following sequence of
operations:

Set the diagonal to D, where D may be input or
computed according to MODE, COND, DMAX, and SYM
as described below.

Generate a matrix with the appropriate band structure, by one
of two methods:

Method A:
Generate a dense M x N matrix by multiplying D on the left
and the right by random unitary matrices, then:

Reduce the bandwidth according to KL and KU, using
Householder transformations.

Method B:
Convert the bandwidth-0 (i.e., diagonal) matrix to a
bandwidth-1 matrix using Givens rotations, "chasing"
out-of-band elements back, much as in QR; then convert
the bandwidth-1 to a bandwidth-2 matrix, etc.  Note
that for reasonably small bandwidths (relative to M and
N) this requires less storage, as a dense matrix is not
generated.  Also, for hermitian or symmetric matrices,
only one triangle is generated.

Method A is chosen if the bandwidth is a large fraction of the
order of the matrix, and LDA is at least M (so a dense
matrix can be stored.)  Method B is chosen if the bandwidth
is small (< 1/2 N for hermitian or symmetric, < .3 N+M for
non-symmetric), or LDA is less than M and not less than the
bandwidth.

Pack the matrix if desired. Options specified by PACK are:
no packing
zero out upper half (if hermitian)
zero out lower half (if hermitian)
store the upper half columnwise (if hermitian or upper
triangular)
store the lower half columnwise (if hermitian or lower
triangular)
store the lower triangle in banded format (if hermitian or
lower triangular)
store the upper triangle in banded format (if hermitian or
upper triangular)
store the entire matrix in banded format
If Method B is chosen, and band format is specified, then the
matrix will be generated in the band format, so no repacking
will be necessary.```
Parameters:
 [in] M ``` M is INTEGER The number of rows of A. Not modified.``` [in] N ``` N is INTEGER The number of columns of A. N must equal M if the matrix is symmetric or hermitian (i.e., if SYM is not 'N') Not modified.``` [in] 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.``` [in,out] 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 ZLATMS to continue the same random number sequence. Changed on exit.``` [in] SYM ``` SYM is CHARACTER*1 If SYM='H', the generated matrix is hermitian, with eigenvalues specified by D, COND, MODE, and DMAX; they may be positive, negative, or zero. If SYM='P', the generated matrix is hermitian, with eigenvalues (= singular values) specified by D, COND, MODE, and DMAX; they will not be negative. If SYM='N', the generated matrix is nonsymmetric, with singular values specified by D, COND, MODE, and DMAX; they will not be negative. If SYM='S', the generated matrix is (complex) symmetric, with singular values specified by D, COND, MODE, and DMAX; they will not be negative. Not modified.``` [in,out] D ``` D is DOUBLE PRECISION 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.``` [in] 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='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.``` [in] COND ``` COND is DOUBLE PRECISION On entry, this is used as described under MODE above. If used, it must be >= 1. Not modified.``` [in] DMAX ``` DMAX is DOUBLE PRECISION 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.``` [in] 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 or hermitian. Not modified.``` [in] 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 or hermitian. Not modified.``` [in] PACK ``` PACK is CHARACTER*1 This specifies packing of matrix as follows: 'N' => no packing 'U' => zero out all subdiagonal entries (if symmetric or hermitian) 'L' => zero out all superdiagonal entries (if symmetric or hermitian) 'C' => store the upper triangle columnwise (only if the matrix is symmetric, hermitian, or upper triangular) 'R' => store the lower triangle columnwise (only if the matrix is symmetric, hermitian, or lower triangular) 'B' => store the lower triangle in band storage scheme (only if the matrix is symmetric, hermitian, or lower triangular) 'Q' => store the upper triangle in band storage scheme (only if the matrix is symmetric, hermitian, or upper triangular) 'Z' => store the entire matrix in band storage scheme (pivoting can be provided for by using this option to store A in the trailing rows of the allocated storage) Using these options, the various LAPACK packed and banded storage schemes can be obtained: GB - use 'Z' PB, SB, HB, or TB - use 'B' or 'Q' PP, SP, HB, or TP - use 'C' or 'R' If two calls to ZLATMS differ only in the PACK parameter, they will generate mathematically equivalent matrices. Not modified.``` [in,out] A ``` A is COMPLEX*16 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.``` [in] 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.``` [out] WORK ``` WORK is COMPLEX*16 array, dimension ( 3*MAX( N, M ) ) Workspace. Modified.``` [out] 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 is not 'N' and KU is not equal to KL -12 => PACK illegal string, or PACK='U' or 'L', and SYM='N'; or PACK='C' or 'Q' and SYM='N' and KL is not zero; or PACK='R' or 'B' and SYM='N' and KU is not zero; or PACK='U', 'L', 'C', 'R', 'B', or 'Q', and M is not N. -14 => LDA is less than M, or PACK='Z' and LDA is less than MIN(KU,N-1) + MIN(KL,M-1) + 1. 1 => Error return from DLATM1 2 => Cannot scale to DMAX (max. sing. value is 0) 3 => Error return from ZLAGGE, CLAGHE or CLAGSY```
Date:
November 2011

Definition at line 332 of file zlatms.f.

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 subroutine zlatmt ( integer M, integer N, character DIST, integer, dimension( 4 ) ISEED, character SYM, double precision, dimension( * ) D, integer MODE, double precision COND, double precision DMAX, integer RANK, integer KL, integer KU, character PACK, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) WORK, integer INFO )

ZLATMT

Purpose:
```    ZLATMT generates random matrices with specified singular values
(or hermitian with specified eigenvalues)
for testing LAPACK programs.

ZLATMT operates by applying the following sequence of
operations:

Set the diagonal to D, where D may be input or
computed according to MODE, COND, DMAX, and SYM
as described below.

Generate a matrix with the appropriate band structure, by one
of two methods:

Method A:
Generate a dense M x N matrix by multiplying D on the left
and the right by random unitary matrices, then:

Reduce the bandwidth according to KL and KU, using
Householder transformations.

Method B:
Convert the bandwidth-0 (i.e., diagonal) matrix to a
bandwidth-1 matrix using Givens rotations, "chasing"
out-of-band elements back, much as in QR; then convert
the bandwidth-1 to a bandwidth-2 matrix, etc.  Note
that for reasonably small bandwidths (relative to M and
N) this requires less storage, as a dense matrix is not
generated.  Also, for hermitian or symmetric matrices,
only one triangle is generated.

Method A is chosen if the bandwidth is a large fraction of the
order of the matrix, and LDA is at least M (so a dense
matrix can be stored.)  Method B is chosen if the bandwidth
is small (< 1/2 N for hermitian or symmetric, < .3 N+M for
non-symmetric), or LDA is less than M and not less than the
bandwidth.

Pack the matrix if desired. Options specified by PACK are:
no packing
zero out upper half (if hermitian)
zero out lower half (if hermitian)
store the upper half columnwise (if hermitian or upper
triangular)
store the lower half columnwise (if hermitian or lower
triangular)
store the lower triangle in banded format (if hermitian or
lower triangular)
store the upper triangle in banded format (if hermitian or
upper triangular)
store the entire matrix in banded format
If Method B is chosen, and band format is specified, then the
matrix will be generated in the band format, so no repacking
will be necessary.```
Parameters:
 [in] M ``` M is INTEGER The number of rows of A. Not modified.``` [in] N ``` N is INTEGER The number of columns of A. N must equal M if the matrix is symmetric or hermitian (i.e., if SYM is not 'N') Not modified.``` [in] 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.``` [in,out] 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 ZLATMT to continue the same random number sequence. Changed on exit.``` [in] SYM ``` SYM is CHARACTER*1 If SYM='H', the generated matrix is hermitian, with eigenvalues specified by D, COND, MODE, and DMAX; they may be positive, negative, or zero. If SYM='P', the generated matrix is hermitian, with eigenvalues (= singular values) specified by D, COND, MODE, and DMAX; they will not be negative. If SYM='N', the generated matrix is nonsymmetric, with singular values specified by D, COND, MODE, and DMAX; they will not be negative. If SYM='S', the generated matrix is (complex) symmetric, with singular values specified by D, COND, MODE, and DMAX; they will not be negative. Not modified.``` [in,out] D ``` D is DOUBLE PRECISION 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.``` [in] 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='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.``` [in] COND ``` COND is DOUBLE PRECISION On entry, this is used as described under MODE above. If used, it must be >= 1. Not modified.``` [in] DMAX ``` DMAX is DOUBLE PRECISION 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.``` [in] RANK ``` RANK is INTEGER The rank of matrix to be generated for modes 1,2,3 only. D( RANK+1:N ) = 0. Not modified.``` [in] 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 or hermitian. Not modified.``` [in] 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 or hermitian. Not modified.``` [in] PACK ``` PACK is CHARACTER*1 This specifies packing of matrix as follows: 'N' => no packing 'U' => zero out all subdiagonal entries (if symmetric or hermitian) 'L' => zero out all superdiagonal entries (if symmetric or hermitian) 'C' => store the upper triangle columnwise (only if the matrix is symmetric, hermitian, or upper triangular) 'R' => store the lower triangle columnwise (only if the matrix is symmetric, hermitian, or lower triangular) 'B' => store the lower triangle in band storage scheme (only if the matrix is symmetric, hermitian, or lower triangular) 'Q' => store the upper triangle in band storage scheme (only if the matrix is symmetric, hermitian, or upper triangular) 'Z' => store the entire matrix in band storage scheme (pivoting can be provided for by using this option to store A in the trailing rows of the allocated storage) Using these options, the various LAPACK packed and banded storage schemes can be obtained: GB - use 'Z' PB, SB, HB, or TB - use 'B' or 'Q' PP, SP, HB, or TP - use 'C' or 'R' If two calls to ZLATMT differ only in the PACK parameter, they will generate mathematically equivalent matrices. Not modified.``` [in,out] A ``` A is COMPLEX*16 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.``` [in] 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.``` [out] WORK ``` WORK is COMPLEX*16 array, dimension ( 3*MAX( N, M ) ) Workspace. Modified.``` [out] 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 is not 'N' and KU is not equal to KL -12 => PACK illegal string, or PACK='U' or 'L', and SYM='N'; or PACK='C' or 'Q' and SYM='N' and KL is not zero; or PACK='R' or 'B' and SYM='N' and KU is not zero; or PACK='U', 'L', 'C', 'R', 'B', or 'Q', and M is not N. -14 => LDA is less than M, or PACK='Z' and LDA is less than MIN(KU,N-1) + MIN(KL,M-1) + 1. 1 => Error return from DLATM7 2 => Cannot scale to DMAX (max. sing. value is 0) 3 => Error return from ZLAGGE, ZLAGHE or ZLAGSY```
Date:
November 2011

Definition at line 340 of file zlatmt.f.

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