SCHKBB tests the reduction of a general real rectangular band


 matrix to bidiagonal form.


 SGBBRD factors a general band matrix A as  Q B P* , where * means


 transpose, B is upper bidiagonal, and Q and P are orthogonal;


 SGBBRD can also overwrite a given matrix C with Q* C .


 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 C are generated.


 The problem dimensions are as follows


    A:          M x N


    Q:          M x M


    P:          N x N


    B:          min(M,N) x min(M,N)


    C:          M x NRHS


 For each generated matrix, 4 tests are performed:


 (1)   | A - Q B PT | / ( |A| max(M,N) ulp ), PT = P'


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


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


 (4)   | Y - Q' C | / ( |Y| max(M,NRHS) ulp ), where Y = Q' C.


 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:


 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 )

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).


          If NSIZES is zero, SCHKBB does nothing.  NSIZES must be at


          least zero.

MVAL

          MVAL is INTEGER array, dimension (NSIZES)


          The values of the matrix row dimension M.

NVAL

          NVAL is INTEGER array, dimension (NSIZES)


          The values of the matrix column dimension N.

NWDTHS

          NWDTHS is INTEGER


          The number of bandwidths to use.  If it is zero,


          SCHKBB does nothing.  It must be at least zero.

KK

          KK is INTEGER array, dimension (NWDTHS)


          An array containing the bandwidths to be used for the band


          matrices.  The values must be at least zero.

NTYPES

          NTYPES is INTEGER


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


          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 matrix is in A.  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 in NN a


          matrix of that size and 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' matrix C.


          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 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 SCHKBB to continue the same random number


          sequence.

THRESH

          THRESH is REAL


          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.

NOUNIT

          NOUNIT is INTEGER


          The FORTRAN unit number for printing out error messages


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

A

          A is REAL array, dimension


                            (LDA, max(NN))


          Used to hold the matrix A.

LDA

          LDA is INTEGER


          The leading dimension of A.  It must be at least 1


          and at least max( NN ).

AB

          AB is REAL array, dimension (LDAB, max(NN))


          Used to hold A in band storage format.

LDAB

          LDAB is INTEGER


          The leading dimension of AB.  It must be at least 2 (not 1!)


          and at least max( KK )+1.

BD

          BD is REAL array, dimension (max(NN))


          Used to hold the diagonal of the bidiagonal matrix computed


          by SGBBRD.

BE

          BE is REAL array, dimension (max(NN))


          Used to hold the off-diagonal of the bidiagonal matrix


          computed by SGBBRD.

Q

          Q is REAL array, dimension (LDQ, max(NN))


          Used to hold the orthogonal matrix Q computed by SGBBRD.

LDQ

          LDQ is INTEGER


          The leading dimension of Q.  It must be at least 1


          and at least max( NN ).

P

          P is REAL array, dimension (LDP, max(NN))


          Used to hold the orthogonal matrix P computed by SGBBRD.

LDP

          LDP is INTEGER


          The leading dimension of P.  It must be at least 1


          and at least max( NN ).

C

          C is REAL array, dimension (LDC, max(NN))


          Used to hold the matrix C updated by SGBBRD.

LDC

          LDC is INTEGER


          The leading dimension of U.  It must be at least 1


          and at least max( NN ).

CC

          CC is REAL array, dimension (LDC, max(NN))


          Used to hold a copy of the matrix C.

WORK

          WORK is REAL array, dimension (LWORK)

LWORK

          LWORK is INTEGER


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


          max( LDA+1, max(NN)+1 )*max(NN).

RESULT

          RESULT is REAL array, dimension (4)


          The values computed by the tests described above.


          The values are currently limited to 1/ulp, to avoid


          overflow.

INFO

          INFO is INTEGER


          If 0, then everything ran OK.
-----------------------------------------------------------------------


       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.


       NTESTT          The total number of tests performed so far.


       NMAX            Largest value in NN.


       NMATS           The number of matrices generated so far.


       NERRS           The number of tests which have exceeded THRESH


                       so far.


       COND, IMODE     Values to be passed to the matrix generators.


       ANORM           Norm of A; passed to matrix generators.


       OVFL, UNFL      Overflow and underflow thresholds.


       ULP, ULPINV     Finest relative precision and its inverse.


       RTOVFL, RTUNFL  Square roots of the previous 2 values.


               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) )

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.