ZDRVBD checks the singular value decomposition (SVD) driver ZGESVD,


 ZGESDD, ZGESVJ, ZGEJSV, ZGESVDX, and ZGESVDQ.


 ZGESVD and ZGESDD factors A = U diag(S) VT, where U and VT are


 unitary and diag(S) is diagonal with the entries of the array S on


 its diagonal. The entries of S are the singular values, nonnegative


 and stored in decreasing order.  U and VT can be optionally not


 computed, overwritten on A, or computed partially.


 A is M by N. Let MNMIN = min( M, N ). S has dimension MNMIN.


 U can be M by M or M by MNMIN. VT can be N by N or MNMIN by N.


 When ZDRVBD is called, a number of matrix 'sizes' (M's and N's)


 and a number of matrix 'types' are specified.  For each size (M,N)


 and each type of matrix, and for the minimal workspace as well as


 workspace adequate to permit blocking, an  M x N  matrix 'A' will be


 generated and used to test the SVD routines.  For each matrix, A will


 be factored as A = U diag(S) VT and the following 12 tests computed:


 Test for ZGESVD:


 (1)   | A - U diag(S) VT | / ( |A| max(M,N) ulp )


 (2)   | I - U'U | / ( M ulp )


 (3)   | I - VT VT' | / ( N ulp )


 (4)   S contains MNMIN nonnegative values in decreasing order.


       (Return 0 if true, 1/ULP if false.)


 (5)   | U - Upartial | / ( M ulp ) where Upartial is a partially


       computed U.


 (6)   | VT - VTpartial | / ( N ulp ) where VTpartial is a partially


       computed VT.


 (7)   | S - Spartial | / ( MNMIN ulp |S| ) where Spartial is the


       vector of singular values from the partial SVD


 Test for ZGESDD:


 (8)   | A - U diag(S) VT | / ( |A| max(M,N) ulp )


 (9)   | I - U'U | / ( M ulp )


 (10)  | I - VT VT' | / ( N ulp )


 (11)  S contains MNMIN nonnegative values in decreasing order.


       (Return 0 if true, 1/ULP if false.)


 (12)  | U - Upartial | / ( M ulp ) where Upartial is a partially


       computed U.


 (13)  | VT - VTpartial | / ( N ulp ) where VTpartial is a partially


       computed VT.


 (14)  | S - Spartial | / ( MNMIN ulp |S| ) where Spartial is the


       vector of singular values from the partial SVD


 Test for ZGESVDQ:


 (36)  | A - U diag(S) VT | / ( |A| max(M,N) ulp )


 (37)  | I - U'U | / ( M ulp )


 (38)  | I - VT VT' | / ( N ulp )


 (39)  S contains MNMIN nonnegative values in decreasing order.


       (Return 0 if true, 1/ULP if false.)


 Test for ZGESVJ:


 (15)  | A - U diag(S) VT | / ( |A| max(M,N) ulp )


 (16)  | I - U'U | / ( M ulp )


 (17)  | I - VT VT' | / ( N ulp )


 (18)  S contains MNMIN nonnegative values in decreasing order.


       (Return 0 if true, 1/ULP if false.)


 Test for ZGEJSV:


 (19)  | A - U diag(S) VT | / ( |A| max(M,N) ulp )


 (20)  | I - U'U | / ( M ulp )


 (21)  | I - VT VT' | / ( N ulp )


 (22)  S contains MNMIN nonnegative values in decreasing order.


        (Return 0 if true, 1/ULP if false.)


 Test for ZGESVDX( 'V', 'V', 'A' )/ZGESVDX( 'N', 'N', 'A' )


 (23)  | A - U diag(S) VT | / ( |A| max(M,N) ulp )


 (24)  | I - U'U | / ( M ulp )


 (25)  | I - VT VT' | / ( N ulp )


 (26)  S contains MNMIN nonnegative values in decreasing order.


       (Return 0 if true, 1/ULP if false.)


 (27)  | U - Upartial | / ( M ulp ) where Upartial is a partially


       computed U.


 (28)  | VT - VTpartial | / ( N ulp ) where VTpartial is a partially


       computed VT.


 (29)  | S - Spartial | / ( MNMIN ulp |S| ) where Spartial is the


       vector of singular values from the partial SVD


 Test for ZGESVDX( 'V', 'V', 'I' )


 (30)  | U' A VT''' - diag(S) | / ( |A| max(M,N) ulp )


 (31)  | I - U'U | / ( M ulp )


 (32)  | I - VT VT' | / ( N ulp )


 Test for ZGESVDX( 'V', 'V', 'V' )


 (33)   | U' A VT''' - diag(S) | / ( |A| max(M,N) ulp )


 (34)   | I - U'U | / ( M ulp )


 (35)   | I - VT VT' | / ( N ulp )


 The 'sizes' are specified by the arrays MM(1:NSIZES) and


 NN(1:NSIZES); the value of each element pair (MM(j),NN(j))


 specifies one size.  The 'types' are specified by a logical array


 DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type 'j'


 will be generated.


 Currently, the list of possible types is:


 (1)  The zero matrix.


 (2)  The identity matrix.


 (3)  A matrix of the form  U D V, where U and V are unitary and


      D has evenly spaced entries 1, ..., ULP with random signs


      on the diagonal.


 (4)  Same as (3), but multiplied by the underflow-threshold / ULP.


 (5)  Same as (3), but multiplied by the overflow-threshold * ULP.

NSIZES

          NSIZES is INTEGER


          The number of sizes of matrices to use.  If it is zero,


          ZDRVBD does nothing.  It must be at least zero.

MM

          MM is INTEGER array, dimension (NSIZES)


          An array containing the matrix 'heights' to be used.  For


          each j=1,...,NSIZES, if MM(j) is zero, then MM(j) and NN(j)


          will be ignored.  The MM(j) values must be at least zero.

NN

          NN is INTEGER array, dimension (NSIZES)


          An array containing the matrix 'widths' to be used.  For


          each j=1,...,NSIZES, if NN(j) is zero, then MM(j) and NN(j)


          will be ignored.  The NN(j) values must be at least zero.

NTYPES

          NTYPES is INTEGER


          The number of elements in DOTYPE.   If it is zero, ZDRVBD


          does nothing.  It must be at least zero.  If it is MAXTYP+1


          and NSIZES is 1, then an additional type, MAXTYP+1 is


          defined, which is to use whatever matrices are in A and B.


          This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and


          DOTYPE(MAXTYP+1) is .TRUE. .

DOTYPE

          DOTYPE is LOGICAL array, dimension (NTYPES)


          If DOTYPE(j) is .TRUE., then for each size (m,n), a matrix


          of type j will be generated.  If NTYPES is smaller than the


          maximum number of types defined (PARAMETER MAXTYP), then


          types NTYPES+1 through MAXTYP will not be generated.  If


          NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through


          DOTYPE(NTYPES) will be ignored.

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 ZDRVBD to continue the same random number


          sequence.

THRESH

          THRESH is DOUBLE PRECISION


          A test will count as 'failed' if the 'error', computed as


          described above, exceeds THRESH.  Note that the error


          is scaled to be O(1), so THRESH should be a reasonably


          small multiple of 1, e.g., 10 or 100.  In particular,


          it should not depend on the precision (single vs. double)


          or the size of the matrix.  It must be at least zero.

A

          A is COMPLEX*16 array, dimension (LDA,max(NN))


          Used to hold the matrix whose singular values are to be


          computed.  On exit, A contains the last matrix actually


          used.

LDA

          LDA is INTEGER


          The leading dimension of A.  It must be at


          least 1 and at least max( MM ).

U

          U is COMPLEX*16 array, dimension (LDU,max(MM))


          Used to hold the computed matrix of right singular vectors.


          On exit, U contains the last such vectors actually computed.

LDU

          LDU is INTEGER


          The leading dimension of U.  It must be at


          least 1 and at least max( MM ).

VT

          VT is COMPLEX*16 array, dimension (LDVT,max(NN))


          Used to hold the computed matrix of left singular vectors.


          On exit, VT contains the last such vectors actually computed.

LDVT

          LDVT is INTEGER


          The leading dimension of VT.  It must be at


          least 1 and at least max( NN ).

ASAV

          ASAV is COMPLEX*16 array, dimension (LDA,max(NN))


          Used to hold a different copy of the matrix whose singular


          values are to be computed.  On exit, A contains the last


          matrix actually used.

USAV

          USAV is COMPLEX*16 array, dimension (LDU,max(MM))


          Used to hold a different copy of the computed matrix of


          right singular vectors. On exit, USAV contains the last such


          vectors actually computed.

VTSAV

          VTSAV is COMPLEX*16 array, dimension (LDVT,max(NN))


          Used to hold a different copy of the computed matrix of


          left singular vectors. On exit, VTSAV contains the last such


          vectors actually computed.

S

          S is DOUBLE PRECISION array, dimension (max(min(MM,NN)))


          Contains the computed singular values.

SSAV

          SSAV is DOUBLE PRECISION array, dimension (max(min(MM,NN)))


          Contains another copy of the computed singular values.

E

          E is DOUBLE PRECISION array, dimension (max(min(MM,NN)))


          Workspace for ZGESVD.

WORK

          WORK is COMPLEX*16 array, dimension (LWORK)

LWORK

          LWORK is INTEGER


          The number of entries in WORK.  This must be at least


          MAX(3*MIN(M,N)+MAX(M,N)**2,5*MIN(M,N),3*MAX(M,N)) for all


          pairs  (M,N)=(MM(j),NN(j))

RWORK

          RWORK is DOUBLE PRECISION array,


                      dimension ( 5*max(max(MM,NN)) )

IWORK

          IWORK is INTEGER array, dimension at least 8*min(M,N)

NOUNIT

          NOUNIT is INTEGER


          The FORTRAN unit number for printing out error messages


          (e.g., if a routine returns IINFO not equal to 0.)

INFO

          INFO is INTEGER


          If 0, then everything ran OK.


           -1: NSIZES < 0


           -2: Some MM(j) < 0


           -3: Some NN(j) < 0


           -4: NTYPES < 0


           -7: THRESH < 0


          -10: LDA < 1 or LDA < MMAX, where MMAX is max( MM(j) ).


          -12: LDU < 1 or LDU < MMAX.


          -14: LDVT < 1 or LDVT < NMAX, where NMAX is max( NN(j) ).


          -21: LWORK too small.


          If  ZLATMS, or ZGESVD returns an error code, the


              absolute value of it is returned.

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