TESTING/EIG/schkst2stg.f(3)

NAME
TESTING/EIG/schkst2stg.f

SYNOPSIS

  Functions/Subroutines
subroutine schkst2stg (nsizes, nn, ntypes, dotype, iseed, thresh, nounit, a, lda, ap, sd, se, d1, d2, d3, d4, d5, wa1, wa2, wa3, wr, u, ldu, v, vp, tau, z, work, lwork, iwork, liwork, result, info)
SCHKST2STG

Function/Subroutine Documentation

  subroutine schkst2stg (integer nsizes, integer, dimension( * ) nn, integer ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed, real thresh, integer nounit, real, dimension( lda, * ) a, integer lda, real, dimension( * ) ap, real, dimension( * ) sd, real, dimension( * ) se, real, dimension( * ) d1, real, dimension( * ) d2, real, dimension( * ) d3, real, dimension( * ) d4, real, dimension( * ) d5, real, dimension( * ) wa1, real, dimension( * ) wa2, real, dimension( * ) wa3, real, dimension( * ) wr, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldu, * ) v, real, dimension( * ) vp, real, dimension( * ) tau, real, dimension( ldu, * ) z, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, real, dimension( * ) result, integer info)

SCHKST2STG

Purpose:

     SCHKST2STG  checks the symmetric eigenvalue problem routines
     using the 2-stage reduction techniques. Since the generation
     of Q or the vectors is not available in this release, we only
     compare the eigenvalue resulting when using the 2-stage to the
     one considered as reference using the standard 1-stage reduction
     SSYTRD. For that, we call the standard SSYTRD and compute D1 using
     SSTEQR, then we call the 2-stage SSYTRD_2STAGE with Upper and Lower
     and we compute D2 and D3 using SSTEQR and then we replaced tests
     3 and 4 by tests 11 and 12. test 1 and 2 remain to verify that
     the 1-stage results are OK and can be trusted.
     This testing routine will converge to the SCHKST in the next
     release when vectors and generation of Q will be implemented.
        SSYTRD factors A as  U S U' , where ' means transpose,
        S is symmetric tridiagonal, and U is orthogonal.
        SSYTRD can use either just the lower or just the upper triangle
        of A; SCHKST2STG checks both cases.
        U is represented as a product of Householder
        transformations, whose vectors are stored in the first
        n-1 columns of V, and whose scale factors are in TAU.
        SSPTRD does the same as SSYTRD, except that A and V are stored
        in 'packed' format.
        SORGTR constructs the matrix U from the contents of V and TAU.
        SOPGTR constructs the matrix U from the contents of VP and TAU.
        SSTEQR factors S as  Z D1 Z' , where Z is the orthogonal
        matrix of eigenvectors and D1 is a diagonal matrix with
        the eigenvalues on the diagonal.  D2 is the matrix of
        eigenvalues computed when Z is not computed.
        SSTERF computes D3, the matrix of eigenvalues, by the
        PWK method, which does not yield eigenvectors.
        SPTEQR factors S as  Z4 D4 Z4' , for a
        symmetric positive definite tridiagonal matrix.
        D5 is the matrix of eigenvalues computed when Z is not
        computed.
        SSTEBZ computes selected eigenvalues.  WA1, WA2, and
        WA3 will denote eigenvalues computed to high
        absolute accuracy, with different range options.
        WR will denote eigenvalues computed to high relative
        accuracy.
        SSTEIN computes Y, the eigenvectors of S, given the
        eigenvalues.
        SSTEDC factors S as Z D1 Z' , where Z is the orthogonal
        matrix of eigenvectors and D1 is a diagonal matrix with
        the eigenvalues on the diagonal ('I' option). It may also
        update an input orthogonal matrix, usually the output
        from SSYTRD/SORGTR or SSPTRD/SOPGTR ('V' option). It may
        also just compute eigenvalues ('N' option).
        SSTEMR factors S as Z D1 Z' , where Z is the orthogonal
        matrix of eigenvectors and D1 is a diagonal matrix with
        the eigenvalues on the diagonal ('I' option).  SSTEMR
        uses the Relatively Robust Representation whenever possible.
     When SCHKST2STG is called, a number of matrix 'sizes' ('n's') and a
     number of matrix 'types' are specified.  For each size ('n')
     and each type of matrix, one matrix will be generated and used
     to test the symmetric eigenroutines.  For each matrix, a number
     of tests will be performed:
     (1)     | A - V S V' | / ( |A| n ulp ) SSYTRD( UPLO='U', ... )
     (2)     | I - UV' | / ( n ulp )        SORGTR( UPLO='U', ... )
     (3)     | A - V S V' | / ( |A| n ulp ) SSYTRD( UPLO='L', ... )
             replaced by | D1 - D2 | / ( |D1| ulp ) where D1 is the
             eigenvalue matrix computed using S and D2 is the
             eigenvalue matrix computed using S_2stage the output of
             SSYTRD_2STAGE('N', 'U',....). D1 and D2 are computed
             via SSTEQR('N',...)
     (4)     | I - UV' | / ( n ulp )        SORGTR( UPLO='L', ... )
             replaced by | D1 - D3 | / ( |D1| ulp ) where D1 is the
             eigenvalue matrix computed using S and D3 is the
             eigenvalue matrix computed using S_2stage the output of
             SSYTRD_2STAGE('N', 'L',....). D1 and D3 are computed
             via SSTEQR('N',...)
     (5-8)   Same as 1-4, but for SSPTRD and SOPGTR.
     (9)     | S - Z D Z' | / ( |S| n ulp ) SSTEQR('V',...)
     (10)    | I - ZZ' | / ( n ulp )        SSTEQR('V',...)
     (11)    | D1 - D2 | / ( |D1| ulp )        SSTEQR('N',...)
     (12)    | D1 - D3 | / ( |D1| ulp )        SSTERF
     (13)    0 if the true eigenvalues (computed by sturm count)
             of S are within THRESH of
             those in D1.  2*THRESH if they are not.  (Tested using
             SSTECH)
     For S positive definite,
     (14)    | S - Z4 D4 Z4' | / ( |S| n ulp ) SPTEQR('V',...)
     (15)    | I - Z4 Z4' | / ( n ulp )        SPTEQR('V',...)
     (16)    | D4 - D5 | / ( 100 |D4| ulp )       SPTEQR('N',...)
     When S is also diagonally dominant by the factor gamma < 1,
     (17)    max | D4(i) - WR(i) | / ( |D4(i)| omega ) ,
              i
             omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
                                                  SSTEBZ( 'A', 'E', ...)
     (18)    | WA1 - D3 | / ( |D3| ulp )          SSTEBZ( 'A', 'E', ...)
     (19)    ( max { min | WA2(i)-WA3(j) | } +
                i     j
               max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
                i     j
                                                  SSTEBZ( 'I', 'E', ...)
     (20)    | S - Y WA1 Y' | / ( |S| n ulp )  SSTEBZ, SSTEIN
     (21)    | I - Y Y' | / ( n ulp )          SSTEBZ, SSTEIN
     (22)    | S - Z D Z' | / ( |S| n ulp )    SSTEDC('I')
     (23)    | I - ZZ' | / ( n ulp )           SSTEDC('I')
     (24)    | S - Z D Z' | / ( |S| n ulp )    SSTEDC('V')
     (25)    | I - ZZ' | / ( n ulp )           SSTEDC('V')
     (26)    | D1 - D2 | / ( |D1| ulp )           SSTEDC('V') and
                                                  SSTEDC('N')
     Test 27 is disabled at the moment because SSTEMR does not
     guarantee high relatvie accuracy.
     (27)    max | D6(i) - WR(i) | / ( |D6(i)| omega ) ,
              i
             omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
                                                  SSTEMR('V', 'A')
     (28)    max | D6(i) - WR(i) | / ( |D6(i)| omega ) ,
              i
             omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
                                                  SSTEMR('V', 'I')
     Tests 29 through 34 are disable at present because SSTEMR
     does not handle partial spectrum requests.
     (29)    | S - Z D Z' | / ( |S| n ulp )    SSTEMR('V', 'I')
     (30)    | I - ZZ' | / ( n ulp )           SSTEMR('V', 'I')
     (31)    ( max { min | WA2(i)-WA3(j) | } +
                i     j
               max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
                i     j
             SSTEMR('N', 'I') vs. SSTEMR('V', 'I')
     (32)    | S - Z D Z' | / ( |S| n ulp )    SSTEMR('V', 'V')
     (33)    | I - ZZ' | / ( n ulp )           SSTEMR('V', 'V')
     (34)    ( max { min | WA2(i)-WA3(j) | } +
                i     j
               max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
                i     j
             SSTEMR('N', 'V') vs. SSTEMR('V', 'V')
     (35)    | S - Z D Z' | / ( |S| n ulp )    SSTEMR('V', 'A')
     (36)    | I - ZZ' | / ( n ulp )           SSTEMR('V', 'A')
     (37)    ( max { min | WA2(i)-WA3(j) | } +
                i     j
               max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
                i     j
             SSTEMR('N', 'A') vs. SSTEMR('V', 'A')
     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 )
     (16) Same as (8), but diagonal elements are all positive.
     (17) Same as (9), but diagonal elements are all positive.
     (18) Same as (10), but diagonal elements are all positive.
     (19) Same as (16), but multiplied by SQRT( overflow threshold )
     (20) Same as (16), but multiplied by SQRT( underflow threshold )
     (21) A diagonally dominant tridiagonal matrix with geometrically
          spaced diagonal entries 1, ..., ULP.

Parameters
    NSIZES

              NSIZES is INTEGER
              The number of sizes of matrices to use.  If it is zero,
              SCHKST2STG 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.

    NTYPES

              NTYPES is INTEGER
              The number of elements in DOTYPE.   If it is zero, SCHKST2STG
              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 SCHKST2STG 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 of
                                      dimension ( LDA , max(NN) )
              Used to hold the matrix whose eigenvalues are to be
              computed.  On exit, A contains the last matrix actually
              used.

    LDA

              LDA is INTEGER
              The leading dimension of A.  It must be at
              least 1 and at least max( NN ).

    AP

              AP is REAL array of
                          dimension( max(NN)*max(NN+1)/2 )
              The matrix A stored in packed format.

    SD

              SD is REAL array of
                                 dimension( max(NN) )
              The diagonal of the tridiagonal matrix computed by SSYTRD.
              On exit, SD and SE contain the tridiagonal form of the
              matrix in A.

    SE

              SE is REAL array of
                                 dimension( max(NN) )
              The off-diagonal of the tridiagonal matrix computed by
              SSYTRD.  On exit, SD and SE contain the tridiagonal form of
              the matrix in A.

    D1

              D1 is REAL array of
                                 dimension( max(NN) )
              The eigenvalues of A, as computed by SSTEQR simultaneously
              with Z.  On exit, the eigenvalues in D1 correspond with the
              matrix in A.

    D2

              D2 is REAL array of
                                 dimension( max(NN) )
              The eigenvalues of A, as computed by SSTEQR if Z is not
              computed.  On exit, the eigenvalues in D2 correspond with
              the matrix in A.

    D3

              D3 is REAL array of
                                 dimension( max(NN) )
              The eigenvalues of A, as computed by SSTERF.  On exit, the
              eigenvalues in D3 correspond with the matrix in A.

    D4

              D4 is REAL array of
                                 dimension( max(NN) )
              The eigenvalues of A, as computed by SPTEQR(V).
              SPTEQR factors S as  Z4 D4 Z4*
              On exit, the eigenvalues in D4 correspond with the matrix in A.

    D5

              D5 is REAL array of
                                 dimension( max(NN) )
              The eigenvalues of A, as computed by SPTEQR(N)
              when Z is not computed. On exit, the
              eigenvalues in D4 correspond with the matrix in A.

    WA1

              WA1 is REAL array of
                                 dimension( max(NN) )
              All eigenvalues of A, computed to high
              absolute accuracy, with different range options.
              as computed by SSTEBZ.

    WA2

              WA2 is REAL array of
                                 dimension( max(NN) )
              Selected eigenvalues of A, computed to high
              absolute accuracy, with different range options.
              as computed by SSTEBZ.
              Choose random values for IL and IU, and ask for the
              IL-th through IU-th eigenvalues.

    WA3

              WA3 is REAL array of
                                 dimension( max(NN) )
              Selected eigenvalues of A, computed to high
              absolute accuracy, with different range options.
              as computed by SSTEBZ.
              Determine the values VL and VU of the IL-th and IU-th
              eigenvalues and ask for all eigenvalues in this range.

    WR

              WR is REAL array of
                                 dimension( max(NN) )
              All eigenvalues of A, computed to high
              absolute accuracy, with different options.
              as computed by SSTEBZ.

    U

              U is REAL array of
                                 dimension( LDU, max(NN) ).
              The orthogonal matrix computed by SSYTRD + SORGTR.

    LDU

              LDU is INTEGER
              The leading dimension of U, Z, and V.  It must be at least 1
              and at least max( NN ).

    V

              V is REAL array of
                                 dimension( LDU, max(NN) ).
              The Housholder vectors computed by SSYTRD in reducing A to
              tridiagonal form.  The vectors computed with UPLO='U' are
              in the upper triangle, and the vectors computed with UPLO='L'
              are in the lower triangle.  (As described in SSYTRD, the
              sub- and superdiagonal are not set to 1, although the
              true Householder vector has a 1 in that position.  The
              routines that use V, such as SORGTR, set those entries to
              1 before using them, and then restore them later.)

    VP

              VP is REAL array of
                          dimension( max(NN)*max(NN+1)/2 )
              The matrix V stored in packed format.

    TAU

              TAU is REAL array of
                                 dimension( max(NN) )
              The Householder factors computed by SSYTRD in reducing A
              to tridiagonal form.

    Z

              Z is REAL array of
                                 dimension( LDU, max(NN) ).
              The orthogonal matrix of eigenvectors computed by SSTEQR,
              SPTEQR, and SSTEIN.

    WORK

              WORK is REAL array of
                          dimension( LWORK )

    LWORK

              LWORK is INTEGER
              The number of entries in WORK.  This must be at least
              1 + 4 * Nmax + 2 * Nmax * lg Nmax + 3 * Nmax**2
              where Nmax = max( NN(j), 2 ) and lg = log base 2.

    IWORK

              IWORK is INTEGER array,
              Workspace.

    LIWORK

              LIWORK is INTEGER
              The number of entries in IWORK.  This must be at least
                      6 + 6*Nmax + 5 * Nmax * lg Nmax
              where Nmax = max( NN(j), 2 ) and lg = log base 2.

    RESULT

              RESULT is REAL array, dimension (26)
              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.
               -1: NSIZES < 0
               -2: Some NN(j) < 0
               -3: NTYPES < 0
               -5: THRESH < 0
               -9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ).
              -23: LDU < 1 or LDU < NMAX.
              -29: LWORK too small.
              If  SLATMR, SLATMS, SSYTRD, SORGTR, SSTEQR, SSTERF,
                  or SORMC2 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.
           NTESTT          The total number of tests performed so far.
           NBLOCK          Blocksize as returned by ENVIR.
           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) )

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 608 of file schkst2stg.f.

Author

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