SCHKSB tests the reduction of a symmetric band matrix to tridiagonal


 form, used with the symmetric eigenvalue problem.


 SSBTRD factors a symmetric band matrix A as  U S U' , where ' means


 transpose, S is symmetric tridiagonal, and U is orthogonal.


 SSBTRD can use either just the lower or just the upper triangle


 of A; SCHKSB checks both cases.


 When SCHKSB is called, a number of matrix 'sizes' ('n's'), a number


 of bandwidths ('k's'), and a number of matrix 'types' are


 specified.  For each size ('n'), each bandwidth ('k') less than or


 equal to 'n', and each type of matrix, one matrix will be generated


 and used to test the symmetric banded reduction routine.  For each


 matrix, a number of tests will be performed:


 (1)     | A - V S V' | / ( |A| n ulp )  computed by SSBTRD with


                                         UPLO='U'


 (2)     | I - UU' | / ( n ulp )


 (3)     | A - V S V' | / ( |A| n ulp )  computed by SSBTRD with


                                         UPLO='L'


 (4)     | I - UU' | / ( n ulp )


 The 'sizes' are specified by an array NN(1:NSIZES); the value of


 each element 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 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 (4), but multiplied by SQRT( overflow threshold )


 (7)  Same as (4), but multiplied by SQRT( underflow threshold )


 (8)  A matrix of the form  U' D U, where U is orthogonal and


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


      on the diagonal.


 (9)  A matrix of the form  U' D U, where U is orthogonal and


      D has geometrically spaced entries 1, ..., ULP with random


      signs on the diagonal.


 (10) A matrix of the form  U' D U, where U is 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) Symmetric 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 sizes of matrices to use.  If it is zero,


          SCHKSB does nothing.  It must be at least zero.

NN

          NN is INTEGER array, dimension (NSIZES)


          An array containing the sizes to be used for the matrices.


          Zero values will be skipped.  The values must be at least


          zero.

NWDTHS

          NWDTHS is INTEGER


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


          SCHKSB 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, SCHKSB


          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.

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 SCHKSB 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 whose eigenvalues are to be


          computed.

LDA

          LDA is INTEGER


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


          and at least max( KK )+1.

SD

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


          Used to hold the diagonal of the tridiagonal matrix computed


          by SSBTRD.

SE

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


          Used to hold the off-diagonal of the tridiagonal matrix


          computed by SSBTRD.

U

          U is REAL array, dimension (LDU, max(NN))


          Used to hold the orthogonal matrix computed by SSBTRD.

LDU

          LDU is INTEGER


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


          and at least max( NN ).

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.