ZCHKHB2STG.F(3)

📦
version 3.12.1 · 3 versions
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🎀 ZCHKHB2STG(3) — TESTING/EIG/zchkhb2stg.f 💅

okay so bestie, sit down, hydrate, because this one is a whole test driver and she has ✨opinions✨. this is TESTING/EIG/zchkhb2stg.f, manual page section 3, and she is the queen of checking whether your Hermitian band reduction actually did what it said it would. no cap.

🌸 SYNOPSIS 🌸

🦋 Functions/Subroutines

subroutine zchkhb2stg (nsizes, nn, nwdths, kk, ntypes, dotype, iseed,
    thresh, nounit, a, lda, sd, se, d1, d2, d3, u, ldu, work, lwork,
    rwork, result, info)

ZCHKHB2STG. that's her. that's the whole vibe. 🫶

💖 Function/Subroutine Documentation 💖

🔥 subroutine zchkhb2stg

here's the full signature, character for character, because we do not mess with types in this house:

subroutine zchkhb2stg (integer nsizes, integer, dimension( * ) nn, integer
    nwdths, integer, dimension( * ) kk, integer ntypes, logical, dimension(
    * ) dotype, integer, dimension( 4 ) iseed, double precision thresh,
    integer nounit, complex*16, dimension( lda, * ) a, integer lda, double
    precision, dimension( * ) sd, double precision, dimension( * ) se,
    double precision, dimension( * ) d1, double precision, dimension( * )
    d2, double precision, dimension( * ) d3, complex*16, dimension( ldu, * )
    u, integer ldu, complex*16, dimension( * ) work, integer lwork, double
    precision, dimension( * ) rwork, double precision, dimension( * )
    result, integer info)

⭐ Purpose

ZCHKHB2STG tests the reduction of a Hermitian band matrix to tridiagonal from, used with the Hermitian eigenvalue problem. (yes, "from" — she said what she said, we're not editing her 👑)

ZHBTRD factors a Hermitian band matrix A as U S U*, where * means conjugate transpose, S is symmetric tridiagonal, and U is unitary. ZHBTRD can use either just the lower or just the upper triangle of A; ZCHKHB2STG checks both cases. equal opportunity tester 💅

ZHETRD_HB2ST factors a Hermitian band matrix A as U S U*, where * means conjugate transpose, S is symmetric tridiagonal, and U is unitary. ZHETRD_HB2ST can use either just the lower or just the upper triangle of A; ZCHKHB2STG 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. (yes, same line as D2 — that's what the page says, don't come for me 🙈)

When ZCHKHB2STG 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. (some of the variable names ghosted us in this paragraph, we're keeping it real about that 😭)

For each matrix, a number of tests will be performed:

  1. | A - V S V* | / ( |A| n ulp ) — computed by ZHBTRD with UPLO='U'
  2. | I - UU* | / ( n ulp )
  3. | A - V S V* | / ( |A| n ulp ) — computed by ZHBTRD 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 ZHETRD_HB2ST with UPLO='U'
  6. | D1 - D3 | / ( |D1| ulp ) — where D1 is computed by DSBTRD with UPLO='U' and D3 is computed by ZHETRD_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.
  • (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 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

NSIZES — NSIZES is INTEGER. it's the number of sizes of matrices to use. if it's zero, ZCHKHB2STG does nothing (girl's off the clock 😴). 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's zero, ZCHKHB2STG 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's zero, ZCHKHB2STG does nothing. it must be at least zero. here's the spicy part 👀 — 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. (read the room, extra entries 💀)

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 ZCHKHB2STG to continue the same random number sequence. so like, she remembers where you left off 🧚

THRESH — THRESH is DOUBLE PRECISION. 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. (two words went missing here too — we mourn them 🕯️)

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*16 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 DOUBLE PRECISION array, dimension (max(NN)). used to hold the diagonal of the tridiagonal matrix computed by ZHBTRD.

SE — SE is DOUBLE PRECISION array, dimension (max(NN)). used to hold the off-diagonal of the tridiagonal matrix computed by ZHBTRD.

D1 — D1 is DOUBLE PRECISION array, dimension (max(NN)).

D2 — D2 is DOUBLE PRECISION array, dimension (max(NN)).

D3 — D3 is DOUBLE PRECISION array, dimension (max(NN)).

U — U is COMPLEX*16 array, dimension (LDU, max(NN)). used to hold the unitary matrix computed by ZHBTRD.

okay so we're literally not even done with the parameters yet, LDU and the whole rest of the crew are still coming 💀 stay with me —

  • 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 .
  • 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 334 of file zchkhb2stg.f.

🌸 Author

Generated automatically by Doxygen for LAPACK from the source code.

That’s the whole vibe, bestie — served fresh from LAPACK Version 3.12.0 for TESTING/EIG/zchkhb2stg.f(3). 🫶💖🎀

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