TESTING/EIG/zchkhb.f(3)

💖 TESTING/EIG/zchkhb.f(3) — the zchkhb manual page, bestie 💖

okay so this is the manual page for TESTING/EIG/zchkhb.f(3), and it’s literally the home of zchkhb. It’s a Fortran test routine with main character energy: it checks the Hermitian band matrix → tridiagonal reduction used with the Hermitian eigenvalue problem. No cap. 🦋

🌸 NAME 🌸

TESTING/EIG/zchkhb.f — yes bestie, that’s the whole name section. Short, iconic, periodt. 💅

✨ SYNOPSIS ✨

Functions/Subroutines

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

ZCHKHB

🦋 Function/Subroutine Documentation 🦋

subroutine zchkhb (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,
    complex*16, dimension( ldu, * ) u, integer ldu, complex*16, dimension( *
    ) work, integer lwork, double precision, dimension( * ) rwork, double
    precision, dimension( * ) result, integer info)
    ZCHKHB

🎀 Purpose 🎀

Okay so ZCHKHB tests the reduction of a Hermitian band matrix to tridiagonal from, used with the Hermitian eigenvalue problem. (Yes, the source says “from” — I’m not fixing it, bestie. We respect the typo. 💀)

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; ZCHKHB checks both cases. Obsessed with the thoroughness. ✨

The source text has a few literal blank spots in the next bits (the manual page is being a little glitchy, ngl), so I’m keeping those blanks as blanks instead of inventing extra names. 🫶

When ZCHKHB 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 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 )

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. The number of sizes of matrices to use. If it is zero, ZCHKHB 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, ZCHKHB 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, ZCHKHB 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 ZCHKHB to continue the same random number sequence.
• 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.
• 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.
• U — U is COMPLEX*16 array, dimension (LDU, max(NN)). Used to hold the unitary matrix computed by ZHBTRD.
• LDU — LDU is INTEGER. The leading dimension of U. It must be at least 1 and at least max( NN ).
• WORK — WORK is COMPLEX*16 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 DOUBLE PRECISION array.
• RESULT — RESULT is DOUBLE PRECISION 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 🧚

The source also has the little underline aesthetic ---- ----- --------- --- ----------, and then it lists these local cuties — exact names, exact details:

• 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 source says “The following four arrays decode JTYPE:” and then gives this trio:

• 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 295 of file zchkhb.f.

🌈 AUTHOR 🌈

okay so real talk, bestie — nobody hand-wrote this one. This page was generated automatically by Doxygen for LAPACK from the source code. 💅 That's the whole author section. A robot did it. And honestly? It ate. ✨

No mysterious human maintainer to @, no cap — just Doxygen doing its little automated thing and giving us this absolute unit of a reference page. 🫶

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and that's a wrap on TESTING/EIG/zchkhb.f(3)! 🎀 if you scrolled all the way down here after the signature and all those arguments, ily, you're the main character fr fr. 💜 This beauty ships with LAPACK Version 3.12.0, straight from the source code, generated by Doxygen — so go forth and test those Hermitian band matrices, icon. 🦋🌸👑