DCHKSB(3)

💖 NAME

Bestie, this is DCHKSB(3) from the Library Functions Manual — a LAPACK testing file, and its whole government name is:

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

That's the file. That's where it lives. Slay. 💅

real talk before we start: the typesetting in the source ate a few variable names, so wherever the original had a blank I'm showing you ___ instead. I'm not about to invent names for you, that's not the vibe 🙈

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✨ SYNOPSIS ✨

🦋 Functions/Subroutines

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

That's the whole cast list, no cap. 🫶

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🎀 FUNCTION/SUBROUTINE DOCUMENTATION 🎀

subroutine DCHKSB (integer nsizes, integer, dimension( * ) nn, integer
    nwdths, integer, dimension( * ) kk, integer ntypes, logical, dimension(
    * ) dotype, integer, dimension( 4 ) iseed, double precision thresh,
    integer nounit, double precision, dimension( lda, * ) a, integer lda,
    double precision, dimension( * ) sd, double precision, dimension( * )
    se, double precision, dimension( ldu, * ) u, integer ldu, double
    precision, dimension( * ) work, integer lwork, double precision,
    dimension( * ) result, integer info)
    DCHKSB

Same thing, but with all the types spelled out. Iconic. 👑

🌸 Purpose 🌸

Okay so, DCHKSB tests the reduction of a symmetric band matrix to tridiagonal form — you know, the one you use with the symmetric eigenvalue problem.

DSBTRD factors a symmetric band matrix A as U S U' — and ' means transpose, bestie, in case the apostrophe caught you off guard — where S is symmetric tridiagonal, and U is orthogonal. DSBTRD can use either just the lower or just the upper triangle of A, and DCHKSB checks both cases. Both. She's thorough. 💅

When DCHKSB 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 symmetric banded reduction routine. For each matrix, a number of tests will be performed:

1. | A - V S V' | / ( |A| n ulp ) — computed by DSBTRD with UPLO='U'
2. | I - UU' | / ( n ulp )
3. | A - V S V' | / ( |A| n ulp ) — computed by DSBTRD with UPLO='L'
4. | I - UU' | / ( n ulp )

Four tests, two UPLO flavours, all accounted for. ✨

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 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 )

Fifteen types, all present and correct, we love a complete roster 🌈

🧚 Parameters 🧚

NSIZES

NSIZES is INTEGER. It's the number of sizes of matrices to use. If it's zero, DCHKSB does nothing — literally nothing, she just vibes. 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, DCHKSB 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, DCHKSB does nothing. It must be at least zero. And here's the spicy bit: if it's MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1, is defined, which is to use whatever matrix is in A. That's only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE.. Ik, very specific. That's the whole point 🎀

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. Boundaries, bestie. We respect them 💅

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 — must, don't try her 😭. 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 DCHKSB to continue the same random number sequence. Continuity! Growth! 🌱

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.

so like... do not make THRESH depend on precision or matrix size. The original says it right there and I'm not about to soften it. That's the rule 🙈

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 DOUBLE PRECISION 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. The original literally put an exclamation point on "not 1" and honestly? Correct energy. 🔥

SD

SD is DOUBLE PRECISION array, dimension (max(NN)). Used to hold the diagonal of the tridiagonal matrix computed by DSBTRD.

SE

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

U

U is DOUBLE PRECISION array, dimension (LDU, max(NN)). Used to hold the orthogonal matrix computed by DSBTRD.

LDU

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

WORK

WORK is DOUBLE PRECISION 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). Write that down bestie, the sizing math does not play 😭

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. Safety first, always 🎀

INFO

INFO is INTEGER. If 0, then everything ran OK. That's it. That's the whole condition. Green flag 🟢

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🧚‍♀️ Some Local Variables and Parameters 🧚‍♀️

Decoding the backstage crew for you:

• 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 (yes, the source says four and then lists three — I'm just the messenger 💀):

• 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.

Full credit, all four, no erasing anyone's name 💖

⭐ Definition ⭐

Definition at line 290 of file dchksb.f.

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and bestie, we're not done — there's a whole part 2 coming, so don't close the tab yet 🦋

✨ AUTHOR ✨

okay so plot twist bestie — nobody wrote this page 😭 it was generated automatically by Doxygen for LAPACK from the source code. like, the docs literally manifested themselves out of the code, no human author to thank, no cap. the source code is the main character here and Doxygen is just its hype person 💅

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fr fr though — if you spot something weird on this page, blame the source code and the Doxygen run, not a person 🫶

and that's the whole page, bestie! you made it to the end 🎀 go take a lil victory lap, hydrate, and remember: it's giving documented ✨

LAPACK — Version 3.12.0 🌈👑 and the page itself came from /home/abui...IG/dchksb.f(3) — mwah, slay, periodt 💖🦋