schkbd.f3

!>
!> SCHKBD checks the singular value decomposition (SVD) routines.
!>
!> SGEBRD reduces a real general m by n matrix A to upper or lower
!> bidiagonal form B by an orthogonal transformation:  Q' * A * P = B
!> (or A = Q * B * P').  The matrix B is upper bidiagonal if m >= n
!> and lower bidiagonal if m < n.
!>
!> SORGBR generates the orthogonal matrices Q and P' from SGEBRD.
!> Note that Q and P are not necessarily square.
!>
!> SBDSQR computes the singular value decomposition of the bidiagonal
!> matrix B as B = U S V'.  It is called three times to compute
!>    1)  B = U S1 V', where S1 is the diagonal matrix of singular
!>        values and the columns of the matrices U and V are the left
!>        and right singular vectors, respectively, of B.
!>    2)  Same as 1), but the singular values are stored in S2 and the
!>        singular vectors are not computed.
!>    3)  A = (UQ) S (P'V'), the SVD of the original matrix A.
!> In addition, SBDSQR has an option to apply the left orthogonal matrix
!> U to a matrix X, useful in least squares applications.
!>
!> SBDSDC computes the singular value decomposition of the bidiagonal
!> matrix B as B = U S V' using divide-and-conquer. It is called twice
!> to compute
!>    1) B = U S1 V', where S1 is the diagonal matrix of singular
!>        values and the columns of the matrices U and V are the left
!>        and right singular vectors, respectively, of B.
!>    2) Same as 1), but the singular values are stored in S2 and the
!>        singular vectors are not computed.
!>
!>  SBDSVDX computes the singular value decomposition of the bidiagonal
!>  matrix B as B = U S V' using bisection and inverse iteration. It is
!>  called six times to compute
!>     1) B = U S1 V', RANGE='A', where S1 is the diagonal matrix of singular
!>         values and the columns of the matrices U and V are the left
!>         and right singular vectors, respectively, of B.
!>     2) Same as 1), but the singular values are stored in S2 and the
!>         singular vectors are not computed.
!>     3) B = U S1 V', RANGE='I', with where S1 is the diagonal matrix of singular
!>         values and the columns of the matrices U and V are the left
!>         and right singular vectors, respectively, of B
!>     4) Same as 3), but the singular values are stored in S2 and the
!>         singular vectors are not computed.
!>     5) B = U S1 V', RANGE='V', with where S1 is the diagonal matrix of singular
!>         values and the columns of the matrices U and V are the left
!>         and right singular vectors, respectively, of B
!>     6) Same as 5), but the singular values are stored in S2 and the
!>         singular vectors are not computed.
!>
!> For each pair of matrix dimensions (M,N) and each selected matrix
!> type, an M by N matrix A and an M by NRHS matrix X are generated.
!> The problem dimensions are as follows
!>    A:          M x N
!>    Q:          M x min(M,N) (but M x M if NRHS > 0)
!>    P:          min(M,N) x N
!>    B:          min(M,N) x min(M,N)
!>    U, V:       min(M,N) x min(M,N)
!>    S1, S2      diagonal, order min(M,N)
!>    X:          M x NRHS
!>
!> For each generated matrix, 14 tests are performed:
!>
!> Test SGEBRD and SORGBR
!>
!> (1)   | A - Q B PT | / ( |A| max(M,N) ulp ), PT = P'
!>
!> (2)   | I - Q' Q | / ( M ulp )
!>
!> (3)   | I - PT PT' | / ( N ulp )
!>
!> Test SBDSQR on bidiagonal matrix B
!>
!> (4)   | B - U S1 VT | / ( |B| min(M,N) ulp ), VT = V'
!>
!> (5)   | Y - U Z | / ( |Y| max(min(M,N),k) ulp ), where Y = Q' X
!>                                                  and   Z = U' Y.
!> (6)   | I - U' U | / ( min(M,N) ulp )
!>
!> (7)   | I - VT VT' | / ( min(M,N) ulp )
!>
!> (8)   S1 contains min(M,N) nonnegative values in decreasing order.
!>       (Return 0 if true, 1/ULP if false.)
!>
!> (9)   | S1 - S2 | / ( |S1| ulp ), where S2 is computed without
!>                                   computing U and V.
!>
!> (10)  0 if the true singular values of B are within THRESH of
!>       those in S1.  2*THRESH if they are not.  (Tested using
!>       SSVDCH)
!>
!> Test SBDSQR on matrix A
!>
!> (11)  | A - (QU) S (VT PT) | / ( |A| max(M,N) ulp )
!>
!> (12)  | X - (QU) Z | / ( |X| max(M,k) ulp )
!>
!> (13)  | I - (QU)'(QU) | / ( M ulp )
!>
!> (14)  | I - (VT PT) (PT'VT') | / ( N ulp )
!>
!> Test SBDSDC on bidiagonal matrix B
!>
!> (15)  | B - U S1 VT | / ( |B| min(M,N) ulp ), VT = V'
!>
!> (16)  | I - U' U | / ( min(M,N) ulp )
!>
!> (17)  | I - VT VT' | / ( min(M,N) ulp )
!>
!> (18)  S1 contains min(M,N) nonnegative values in decreasing order.
!>       (Return 0 if true, 1/ULP if false.)
!>
!> (19)  | S1 - S2 | / ( |S1| ulp ), where S2 is computed without
!>                                   computing U and V.
!>  Test SBDSVDX on bidiagonal matrix B
!>
!>  (20)  | B - U S1 VT | / ( |B| min(M,N) ulp ), VT = V'
!>
!>  (21)  | I - U' U | / ( min(M,N) ulp )
!>
!>  (22)  | I - VT VT' | / ( min(M,N) ulp )
!>
!>  (23)  S1 contains min(M,N) nonnegative values in decreasing order.
!>        (Return 0 if true, 1/ULP if false.)
!>
!>  (24)  | S1 - S2 | / ( |S1| ulp ), where S2 is computed without
!>                                    computing U and V.
!>
!>  (25)  | S1 - U' B VT' | / ( |S| n ulp )    SBDSVDX('V', 'I')
!>
!>  (26)  | I - U' U | / ( min(M,N) ulp )
!>
!>  (27)  | I - VT VT' | / ( min(M,N) ulp )
!>
!>  (28)  S1 contains min(M,N) nonnegative values in decreasing order.
!>        (Return 0 if true, 1/ULP if false.)
!>
!>  (29)  | S1 - S2 | / ( |S1| ulp ), where S2 is computed without
!>                                    computing U and V.
!>
!>  (30)  | S1 - U' B VT' | / ( |S1| n ulp )   SBDSVDX('V', 'V')
!>
!>  (31)  | I - U' U | / ( min(M,N) ulp )
!>
!>  (32)  | I - VT VT' | / ( min(M,N) ulp )
!>
!>  (33)  S1 contains min(M,N) nonnegative values in decreasing order.
!>        (Return 0 if true, 1/ULP if false.)
!>
!>  (34)  | S1 - S2 | / ( |S1| ulp ), where S2 is computed without
!>                                    computing U and V.
!>
!> The possible matrix types are
!>
!> (1)  The zero matrix.
!> (2)  The identity matrix.
!>
!> (3)  A diagonal matrix with evenly spaced entries
!>      1, ..., ULP  and random signs.
!>      (ULP = (first number larger than 1) - 1 )
!> (4)  A diagonal matrix with geometrically spaced entries
!>      1, ..., ULP  and random signs.
!> (5)  A diagonal matrix with  entries 1, ULP, ..., ULP
!>      and random signs.
!>
!> (6)  Same as (3), but multiplied by SQRT( overflow threshold )
!> (7)  Same as (3), but multiplied by SQRT( underflow threshold )
!>
!> (8)  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.
!>
!> (9)  A matrix of the form  U D V, where U and V are orthogonal and
!>      D has geometrically spaced entries 1, ..., ULP with random
!>      signs on the diagonal.
!>
!> (10) A matrix of the form  U D V, where U and V are orthogonal and
!>      D has  entries 1, ULP,..., ULP with random
!>      signs on the diagonal.
!>
!> (11) Same as (8), but multiplied by SQRT( overflow threshold )
!> (12) Same as (8), but multiplied by SQRT( underflow threshold )
!>
!> (13) Rectangular matrix with random entries chosen from (-1,1).
!> (14) Same as (13), but multiplied by SQRT( overflow threshold )
!> (15) Same as (13), but multiplied by SQRT( underflow threshold )
!>
!> Special case:
!> (16) A bidiagonal matrix with random entries chosen from a
!>      logarithmic distribution on [ulp^2,ulp^(-2)]  (I.e., each
!>      entry is  e^x, where x is chosen uniformly on
!>      [ 2 log(ulp), -2 log(ulp) ] .)  For *this* type:
!>      (a) SGEBRD is not called to reduce it to bidiagonal form.
!>      (b) the bidiagonal is  min(M,N) x min(M,N); if M<N, the
!>          matrix will be lower bidiagonal, otherwise upper.
!>      (c) only tests 5--8 and 14 are performed.
!>
!> A subset of the full set of matrix types may be selected through
!> the logical array DOTYPE.
!> 

NSIZES

!>          NSIZES is INTEGER
!>          The number of values of M and N contained in the vectors
!>          MVAL and NVAL.  The matrix sizes are used in pairs (M,N).
!> 

MVAL

!>          MVAL is INTEGER array, dimension (NM)
!>          The values of the matrix row dimension M.
!> 

NVAL

!>          NVAL is INTEGER array, dimension (NM)
!>          The values of the matrix column dimension N.
!> 

NTYPES

!>          NTYPES is INTEGER
!>          The number of elements in DOTYPE.   If it is zero, SCHKBD
!>          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.
!> 

NRHS

!>          NRHS is INTEGER
!>          The number of columns in the  matrices X, Y,
!>          and Z, used in testing SBDSQR.  If NRHS = 0, then the
!>          operations on the right-hand side will not be tested.
!>          NRHS must be at least 0.
!> 

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 values of ISEED are changed on exit, and can be
!>          used in the next call to SCHKBD 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.  To have
!>          every test ratio printed, use THRESH = 0.  Note that the
!>          expected value of the test ratios is O(1), so THRESH should
!>          be a reasonably small multiple of 1, e.g., 10 or 100.
!> 

A

!>          A is REAL array, dimension (LDA,NMAX)
!>          where NMAX is the maximum value of N in NVAL.
!> 

LDA

!>          LDA is INTEGER
!>          The leading dimension of the array A.  LDA >= max(1,MMAX),
!>          where MMAX is the maximum value of M in MVAL.
!> 

BD

!>          BD is REAL array, dimension
!>                      (max(min(MVAL(j),NVAL(j))))
!> 

BE

!>          BE is REAL array, dimension
!>                      (max(min(MVAL(j),NVAL(j))))
!> 

S1

!>          S1 is REAL array, dimension
!>                      (max(min(MVAL(j),NVAL(j))))
!> 

S2

!>          S2 is REAL array, dimension
!>                      (max(min(MVAL(j),NVAL(j))))
!> 

X

!>          X is REAL array, dimension (LDX,NRHS)
!> 

LDX

!>          LDX is INTEGER
!>          The leading dimension of the arrays X, Y, and Z.
!>          LDX >= max(1,MMAX)
!> 

Y

!>          Y is REAL array, dimension (LDX,NRHS)
!> 

Z

!>          Z is REAL array, dimension (LDX,NRHS)
!> 

Q

!>          Q is REAL array, dimension (LDQ,MMAX)
!> 

LDQ

!>          LDQ is INTEGER
!>          The leading dimension of the array Q.  LDQ >= max(1,MMAX).
!> 

PT

!>          PT is REAL array, dimension (LDPT,NMAX)
!> 

LDPT

!>          LDPT is INTEGER
!>          The leading dimension of the arrays PT, U, and V.
!>          LDPT >= max(1, max(min(MVAL(j),NVAL(j)))).
!> 

U

!>          U is REAL array, dimension
!>                      (LDPT,max(min(MVAL(j),NVAL(j))))
!> 

VT

!>          VT is REAL array, dimension
!>                      (LDPT,max(min(MVAL(j),NVAL(j))))
!> 

WORK

!>          WORK is REAL array, dimension (LWORK)
!> 

LWORK

!>          LWORK is INTEGER
!>          The number of entries in WORK.  This must be at least
!>          3(M+N) and  M(M + max(M,N,k) + 1) + N*min(M,N)  for all
!>          pairs  (M,N)=(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
!>           -6: NRHS  < 0
!>           -8: THRESH < 0
!>          -11: LDA < 1 or LDA < MMAX, where MMAX is max( MM(j) ).
!>          -17: LDB < 1 or LDB < MMAX.
!>          -21: LDQ < 1 or LDQ < MMAX.
!>          -23: LDPT< 1 or LDPT< MNMAX.
!>          -27: LWORK too small.
!>          If  SLATMR, SLATMS, SGEBRD, SORGBR, or SBDSQR,
!>              returns an error code, the
!>              absolute value of it is returned.
!>
!>-----------------------------------------------------------------------
!>
!>     Some Local Variables and Parameters:
!>     ---- ----- --------- --- ----------
!>
!>     ZERO, ONE       Real 0 and 1.
!>     MAXTYP          The number of types defined.
!>     NTEST           The number of tests performed, or which can
!>                     be performed so far, for the current matrix.
!>     MMAX            Largest value in NN.
!>     NMAX            Largest value in NN.
!>     MNMIN           min(MM(j), NN(j)) (the dimension of the bidiagonal
!>                     matrix.)
!>     MNMAX           The maximum value of MNMIN for j=1,...,NSIZES.
!>     NFAIL           The number of tests which have exceeded THRESH
!>     COND, IMODE     Values to be passed to the matrix generators.
!>     ANORM           Norm of A; passed to matrix generators.
!>
!>     OVFL, UNFL      Overflow and underflow thresholds.
!>     RTOVFL, RTUNFL  Square roots of the previous 2 values.
!>     ULP, ULPINV     Finest relative precision and its inverse.
!>
!>             The following four arrays decode JTYPE:
!>     KTYPE(j)        The general type (1-10) for type .
!>     KMODE(j)        The MODE value to be passed to the matrix
!>                     generator for type .
!>     KMAGN(j)        The order of magnitude ( O(1),
!>                     O(overflow^(1/2) ), O(underflow^(1/2) )
!> 

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