CCHKHB2STG(3)

📦 version 3.12.0, in opensuse-leap 16.0

💅 CCHKHB2STG(3) 💅

Library Functions Manual

Okay bestie, buckle up — this is the LAPACK test-driver page for CCHKHB2STG, and we are gonna make a Fortran subroutine feel like a group chat. Every flag, dimension, and slightly cursed blank spot is staying exactly where it belongs. No cap. 💖

🎀 NAME 🎀

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/TESTING/EIG/cchkhb2stg.f

That's the whole NAME entry — it's giving "file path as a name" energy, and honestly? Iconic. 💀

🦋 SYNOPSIS 🦋

🧚 Functions/Subroutines 🧚

       subroutine CCHKHB2STG (nsizes, nn, nwdths, kk, ntypes, dotype, iseed,
           thresh, nounit, a, lda, sd, se, d1, d2, d3, u, ldu, work, lwork,
           rwork, result, info)
       CCHKHB2STG

🫶 Function/Subroutine Documentation 🫶

Here's the full typed-out signature, bestie — yes, it's a whole paragraph and yes, we're respecting every character:

   subroutine CCHKHB2STG (integer nsizes, integer, dimension( * ) nn, integer
       nwdths, integer, dimension( * ) kk, integer ntypes, logical, dimension(
       * ) dotype, integer, dimension( 4 ) iseed, real thresh, integer nounit,
       complex, dimension( lda, * ) a, integer lda, real, dimension( * ) sd,
       real, dimension( * ) se, real, dimension( * ) d1, real, dimension( * )
       d2, real, dimension( * ) d3, complex, dimension( ldu, * ) u, integer
       ldu, complex, dimension( * ) work, integer lwork, real, dimension( * )
       rwork, real, dimension( * ) result, integer info)
       CCHKHB2STG

CCHKHB2STG

🌟 Purpose 🌟

CCHKHB2STG tests the reduction of a Hermitian band matrix to tridiagonal from, used with the Hermitian eigenvalue problem. (Yes, "from" is literally what the page says — I'm not editing the lore. 💀)

CHBTRD factors a Hermitian band matrix A as U S U*, where * means conjugate transpose, S is symmetric tridiagonal, and U is unitary. CHBTRD can use either just the lower or just the upper triangle of A; CCHKHB2STG checks both cases. Both triangles get their moment. 💅

CHETRD_HB2ST factors a Hermitian band matrix A as U S U*, where * means conjugate transpose, S is symmetric tridiagonal, and U is unitary. CHETRD_HB2ST can use either just the lower or just the upper triangle of A; CCHKHB2STG checks both cases.

DSTEQR factors S as Z D1 Z'. D1 is the matrix of eigenvalues computed when Z is not computed and from the S resulting of DSBTRD (used as reference for DSYTRD_SB2ST). D2 is the matrix of eigenvalues computed when Z is not computed and from the S resulting of DSYTRD_SB2ST. D3 is the matrix of eigenvalues computed when Z is not computed and from the S resulting of DSYTRD_SB2ST.

When CCHKHB2STG is called, a number of matrix (), a number of bandwidths (), and a number of matrix are specified. For each size (), each bandwidth () less than or equal to , and each type of matrix, one matrix will be generated and used to test the hermitian banded reduction routine. For each matrix, a number of tests will be performed:

(1)     | A - V S V* | / ( |A| n ulp )  computed by CHBTRD with
                                        UPLO='U'

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

(3)     | A - V S V* | / ( |A| n ulp )  computed by CHBTRD with
                                        UPLO='L'

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

(5)     | D1 - D2 | / ( |D1| ulp )      where D1 is computed by
                                        DSBTRD with UPLO='U' and
                                        D2 is computed by
                                        CHETRD_HB2ST with UPLO='U'

(6)     | D1 - D3 | / ( |D1| ulp )      where D1 is computed by
                                        DSBTRD with UPLO='U' and
                                        D3 is computed by
                                        CHETRD_HB2ST with UPLO='L'

The are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type will be generated. Currently, the list of possible types is:

  • (1) The zero matrix. — minimalism, we love her.
  • (2) The identity matrix. — main character energy.
  • (3) A diagonal matrix with evenly spaced entries 1, ..., ULP and random signs. (ULP = (first number larger than 1) - 1) — the math is right there.
  • (4) A diagonal matrix with geometrically spaced entries 1, ..., ULP and random signs. — geometry era.
  • (5) A diagonal matrix with entries 1, ULP, ..., ULP and random signs.
  • (6) Same as (4), but multiplied by SQRT( overflow threshold ). — spicy scaling.
  • (7) Same as (4), but multiplied by SQRT( underflow threshold ). — tiny scaling.
  • (8) A matrix of the form U* D U, where U is unitary 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 unitary 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 unitary 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) Hermitian 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 🌈

Okay bestie, parameter roll call — every single one, no skips:

  • NSIZES — NSIZES is INTEGER. The number of sizes of matrices to use. If it is zero, CCHKHB2STG 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, CCHKHB2STG 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, CCHKHB2STG 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 CCHKHB2STG to continue the same random number sequence.
  • THRESH — THRESH is REAL. A test will count as if the , 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 COMPLEX 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 CHBTRD.
  • SE — SE is REAL array, dimension (max(NN)). Used to hold the off-diagonal of the tridiagonal matrix computed by CHBTRD.
  • D1 — D1 is REAL array, dimension (max(NN)). Used store eigenvalues resulting from the tridiagonal form using the DSBTRD.
  • D2 — D2 is REAL array, dimension (max(NN)).
  • D3 — D3 is REAL array, dimension (max(NN)).
  • U — U is COMPLEX array, dimension (LDU, max(NN)). Used to hold the unitary matrix computed by CHBTRD.
  • LDU — LDU is INTEGER. The leading dimension of U. It must be at least 1 and at least max( NN ).
  • WORK — WORK is COMPLEX 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).
  • RWORK — RWORK is REAL array.
  • 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 it's 0, then everything ran OK. That's the green flag, bestie. 🟢✨

🧸 Some Local Variables and Parameters

  • ZERO, ONE — real 0 and 1, the iconic little constants.
  • 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 .
  • 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

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

Definition at line 337 of file cchkhb2stg.f.

👑 Author

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