ZCHKBD checks the singular value decomposition (SVD) routines.


 ZGEBRD reduces a complex general m by n matrix A to real upper or


 lower bidiagonal form 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.


 ZUNGBR generates the orthogonal matrices Q and P' from ZGEBRD.


 Note that Q and P are not necessarily square.


 ZBDSQR 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, ZBDSQR has an option to apply the left orthogonal matrix


 U to a matrix X, useful in least squares applications.


 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 ZGEBRD and ZUNGBR


 (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 ZBDSQR 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)   0 if the true singular values of B are within THRESH of


       those in S1.  2*THRESH if they are not.  (Tested using


       DSVDCH)


 (10)  | S1 - S2 | / ( |S1| ulp ), where S2 is computed without


                                   computing U and V.


 Test ZBDSQR 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 )


 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 'clustered' 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 'clustered' 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) ZGEBRD 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, ZCHKBD


          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 'right-hand side' matrices X, Y,


          and Z, used in testing ZBDSQR.  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 ZCHKBD to continue the same random


          number sequence.

THRESH

          THRESH is DOUBLE PRECISION


          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 COMPLEX*16 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 DOUBLE PRECISION array, dimension


                      (max(min(MVAL(j),NVAL(j))))

BE

          BE is DOUBLE PRECISION array, dimension


                      (max(min(MVAL(j),NVAL(j))))

S1

          S1 is DOUBLE PRECISION array, dimension


                      (max(min(MVAL(j),NVAL(j))))

S2

          S2 is DOUBLE PRECISION array, dimension


                      (max(min(MVAL(j),NVAL(j))))

X

          X is COMPLEX*16 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 COMPLEX*16 array, dimension (LDX,NRHS)

Z

          Z is COMPLEX*16 array, dimension (LDX,NRHS)

Q

          Q is COMPLEX*16 array, dimension (LDQ,MMAX)

LDQ

          LDQ is INTEGER


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

PT

          PT is COMPLEX*16 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 COMPLEX*16 array, dimension


                      (LDPT,max(min(MVAL(j),NVAL(j))))

VT

          VT is COMPLEX*16 array, dimension


                      (LDPT,max(min(MVAL(j),NVAL(j))))

WORK

          WORK is COMPLEX*16 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))

RWORK

          RWORK is DOUBLE PRECISION array, dimension


                      (5*max(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: LDP < 1 or LDP < MNMAX.


          -27: LWORK too small.


          If  ZLATMR, CLATMS, ZGEBRD, ZUNGBR, or ZBDSQR,


              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 'j'.


     KMODE(j)        The MODE value to be passed to the matrix


                     generator for type 'j'.


     KMAGN(j)        The order of magnitude ( O(1),


                     O(overflow^(1/2) ), O(underflow^(1/2) )

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