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.)

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.

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