SCHKSB2STG(3)

📦 version 3.12.0, in opensuse-leap 16.0

💅 SCHKSB2STG(3) — Library Functions Manual 💅

okay bestie, welcome to the schksb2stg.f(3) page — yes it’s a whole LAPACK test routine and it’s lowkey iconic.

✨ NAME ✨

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/TESTING/EIG/schksb2stg.f

That’s the file. That’s the whole NAME section. She’s minimal and mysterious.

🌸 SYNOPSIS 🌸

🎀 Functions/Subroutines 🎀

  • subroutine SCHKSB2STG (nsizes, nn, nwdths, kk, ntypes, dotype, iseed, thresh, nounit, a, lda, sd, se, d1, d2, d3, u, ldu, work, lwork, result, info) — the main character.
  • SCHKSB2STG — yes she gets a second billing too.
subroutine SCHKSB2STG (nsizes, nn, nwdths, kk, ntypes, dotype, iseed,
    thresh, nounit, a, lda, sd, se, d1, d2, d3, u, ldu, work, lwork,
    result, info)

🦋 Function/Subroutine Documentation 🦋

💖 subroutine SCHKSB2STG 💖

subroutine SCHKSB2STG (integer nsizes, integer, dimension( * ) nn, integer
    nwdths, integer, dimension( * ) kk, integer ntypes, logical, dimension(
    * ) dotype, integer, dimension( 4 ) iseed, real thresh, integer nounit,
    real, dimension( lda, * ) a, integer lda, real, dimension( * ) sd, real,
    dimension( * ) se, real, dimension( * ) d1, real, dimension( * ) d2,
    real, dimension( * ) d3, real, dimension( ldu, * ) u, integer ldu, real,
    dimension( * ) work, integer lwork, real, dimension( * ) result, integer
    info)

SCHKSB2STG

🍓 Purpose 🍓

bestie, here’s the whole point: SCHKSB2STG 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; SCHKSB2STG checks both cases.

SSYTRD_SB2ST factors a symmetric band matrix A as U S U', where ' means transpose, S is symmetric tridiagonal, and U is orthogonal. SSYTRD_SB2ST can use either just the lower or just the upper triangle of A; SCHKSB2STG checks both cases.

SSTEQR factors S as Z D1 Z'. D1 is the matrix of eigenvalues computed when Z is not computed and from the S resulting of SSBTRD (used as reference for SSYTRD_SB2ST). D2 is the matrix of eigenvalues computed when Z is not computed and from the S resulting of SSYTRD_SB2ST. D3 is the matrix of eigenvalues computed when Z is not computed and from the S resulting of SSYTRD_SB2ST.

When SCHKSB2STG is called, a number of matrix sizes (NSIZES), a number of bandwidths (NWDTHS), and a number of matrix types (NTYPES) are specified. For each size (NN entry), each bandwidth (KK entry) less than or equal to that size, 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 )
  5. | D1 - D2 | / ( |D1| ulp ) where D1 is computed by SSBTRD with UPLO='U' and D2 is computed by SSYTRD_SB2ST with UPLO='U'
  6. | D1 - D3 | / ( |D1| ulp ) where D1 is computed by SSBTRD with UPLO='U' and D3 is computed by SSYTRD_SB2ST with UPLO='L'

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 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 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 )

🌈 Parameters 🌈

  • NSIZES — INTEGER. The number of sizes of matrices to use. If it is zero, SCHKSB2STG does nothing. It must be at least zero.
  • NN — 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 — INTEGER. The number of bandwidths to use. If it is zero, SCHKSB2STG does nothing. It must be at least zero.
  • KK — INTEGER array, dimension (NWDTHS). An array containing the bandwidths to be used for the band matrices. The values must be at least zero.
  • NTYPES — INTEGER. The number of elements in DOTYPE. If it is zero, SCHKSB2STG 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 — 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 — 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 SCHKSB2STG to continue the same random number sequence.
  • THRESH — 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 — INTEGER. The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.)
  • A — REAL array, dimension (LDA, max(NN)). Used to hold the matrix whose eigenvalues are to be computed.
  • LDA — INTEGER. The leading dimension of A. It must be at least 2 (not 1!) and at least max( KK )+1.
Not 1, bestie. The original is literally yelling “not 1!” and honestly same.
  • SD — REAL array, dimension (max(NN)). Used to hold the diagonal of the tridiagonal matrix computed by SSBTRD.
  • SE — REAL array, dimension (max(NN)). Used to hold the off-diagonal of the tridiagonal matrix computed by SSBTRD.
  • D1 — REAL array, dimension (max(NN)).
  • D2 — REAL array, dimension (max(NN)).
  • D3 — REAL array, dimension (max(NN)).
  • U — REAL array, dimension (LDU, max(NN)). Used to hold the orthogonal matrix computed by SSBTRD.
  • LDU — INTEGER. The leading dimension of U. It must be at least 1 and at least max( NN ).
  • WORK — REAL array, dimension (LWORK).
  • LWORK — INTEGER. The number of entries in WORK. This must be at least max( LDA+1, max(NN)+1 )*max(NN).
  • RESULT — REAL array, dimension (4). The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow.
  • INFO — the description is cut off in this chunk; part 2 has the rest, bestie ✨

One more parameter before we get into the local-variable lore 👀

  • INFO is INTEGER — if it’s 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 (the page says four, and we are simply relaying the message 😭):

  • 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) ) ).

Author 🌟

Shoutout to the institutions behind this masterpiece:

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

Definition at line 329 of file schksb2stg.f.

Author (Doxygen’s lil signature) 🎀

Generated automatically by Doxygen for LAPACK from the source code.

Okay that’s the whole page, bestie — the footer says LAPACK Version 3.12.0, source /home/abui...chksb2stg.f(3). Stay iconic 💅✨

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