SDRVBD checks the singular value decomposition (SVD) drivers


 SGESVD, SGESDD, SGESVDQ, SGESVJ, SGEJSV, and DGESVDX.


 Both SGESVD and SGESDD factor A = U diag(S) VT, where U and VT are


 orthogonal 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 SDRVBD 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 SGESVD:


 (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 SGESDD:


 (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 SGESVDQ:


 (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 SGESVJ:


 (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 SGEJSV:


 (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 SGESVDX( 'V', 'V', 'A' )/SGESVDX( '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 SGESVDX( '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 SGESVDX( '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 orthogonal 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 matrix sizes (M,N) contained in the vectors


          MM and NN.

MM

          MM is INTEGER array, dimension (NSIZES)


          The values of the matrix row dimension M.

NN

          NN is INTEGER array, dimension (NSIZES)


          The values of the matrix column dimension N.

NTYPES

          NTYPES is INTEGER


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


          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, 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.


          On exit, ISEED is changed and can be used in the next call to


          SDRVBD to continue the same random number sequence.

THRESH

          THRESH is REAL


          The threshold value for the test ratios.  A result is


          included in the output file if RESULT >= THRESH.  The test


          ratios are scaled to be O(1), so THRESH should be a small


          multiple of 1, e.g., 10 or 100.  To have every test ratio


          printed, use THRESH = 0.

A

          A is REAL array, dimension (LDA,NMAX)


          where NMAX is the maximum value of N in NN.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,MMAX),


          where MMAX is the maximum value of M in MM.

U

          U is REAL array, dimension (LDU,MMAX)

LDU

          LDU is INTEGER


          The leading dimension of the array U.  LDU >= max(1,MMAX).

VT

          VT is REAL array, dimension (LDVT,NMAX)

LDVT

          LDVT is INTEGER


          The leading dimension of the array VT.  LDVT >= max(1,NMAX).

ASAV

          ASAV is REAL array, dimension (LDA,NMAX)

USAV

          USAV is REAL array, dimension (LDU,MMAX)

VTSAV

          VTSAV is REAL array, dimension (LDVT,NMAX)

S

          S is REAL array, dimension


                      (max(min(MM,NN)))

SSAV

          SSAV is REAL array, dimension


                      (max(min(MM,NN)))

E

          E is REAL array, dimension


                      (max(min(MM,NN)))

WORK

          WORK is REAL array, dimension (LWORK)

LWORK

          LWORK is INTEGER


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


          max(3*MN+MX,5*MN-4)+2*MN**2 for all pairs


          pairs  (MN,MX)=( min(MM(j),NN(j), max(MM(j),NN(j)) )

IWORK

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

NOUT

          NOUT 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  SLATMS, or SGESVD returns an error code, the


              absolute value of it is returned.

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