ZCHKBB(3)

📦 version 3.12.0, in opensuse-leap 16.0

💅 ZCHKBB(3) — Library Functions Manual 💅

okay bestie, this is part 1 of 2 of the ZCHKBB saga ✨ it's from the Library Functions Manual, and we're keeping every last parameter, formula, and little weirdness intact. No cap. 🫶

🎀 NAME 🎀

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

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine ZCHKBB (nsizes, mval, nval, nwdths, kk, ntypes, dotype, nrhs,
    iseed, thresh, nounit, a, lda, ab, ldab, bd, be, q, ldq, p, ldp, c,
    ldc, cc, work, lwork, rwork, result, info)
    ZCHKBB

💖 Function/Subroutine Documentation 💖

subroutine ZCHKBB (integer nsizes, integer, dimension( * ) mval, integer,
    dimension( * ) nval, integer nwdths, integer, dimension( * ) kk, integer
    ntypes, logical, dimension( * ) dotype, integer nrhs, integer,
    dimension( 4 ) iseed, double precision thresh, integer nounit,
    complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension(
    ldab, * ) ab, integer ldab, double precision, dimension( * ) bd, double
    precision, dimension( * ) be, complex*16, dimension( ldq, * ) q, integer
    ldq, complex*16, dimension( ldp, * ) p, integer ldp, complex*16,
    dimension( ldc, * ) c, integer ldc, complex*16, dimension( ldc, * ) cc,
    complex*16, dimension( * ) work, integer lwork, double precision,
    dimension( * ) rwork, double precision, dimension( * ) result, integer
    info)
    ZCHKBB

🌸 Purpose 🌸

okay so here's the main-character job: ZCHKBB tests the reduction of a general complex rectangular band matrix to real bidiagonal form. 👑

ZGBBRD factors a general band matrix A as Q B P* — where * means conjugate transpose, B is upper bidiagonal, and Q and P are unitary. ZGBBRD can also overwrite a given matrix C with Q* C.

For each pair of matrix dimensions (M,N) and each selected matrix type, an M by N matrix A and an M by NRHS matrix C are generated. The problem dimensions are as follows:

A:          M x N
Q:          M x M
P:          N x N
B:          min(M,N) x min(M,N)
C:          M x NRHS

For each generated matrix, 4 tests are performed:

(1)   | A - Q B PT | / ( |A| max(M,N) ulp ), PT = P'

(2)   | I - Q' Q | / ( M ulp )

(3)   | I - PT PT' | / ( N ulp )

(4)   | Y - Q' C | / ( |Y| max(M,NRHS) ulp ), where Y = Q' C.

The matrix types 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: The possible matrix types are:

  1. Type (1) — The zero matrix.
  2. Type (2) — The identity matrix.
  3. Type (3) — A diagonal matrix with evenly spaced entries 1, ..., ULP and random signs. (ULP = (first number larger than 1) - 1)
  4. Type (4) — A diagonal matrix with geometrically spaced entries 1, ..., ULP and random signs.
  5. Type (5) — A diagonal matrix with entries 1, ULP, ..., ULP and random signs.
  6. Type (6) — Same as (3), but multiplied by SQRT( overflow threshold ).
  7. Type (7) — Same as (3), but multiplied by SQRT( underflow threshold ).
  8. Type (8) — A matrix of the form U D V, where U and V are orthogonal and D has evenly spaced entries 1, ..., ULP with random signs on the diagonal.
  9. Type (9) — A matrix of the form U D V, where U and V are orthogonal and D has geometrically spaced entries 1, ..., ULP with random signs on the diagonal.
  10. Type (10) — A matrix of the form U D V, where U and V are orthogonal and D has entries 1, ULP,..., ULP with random signs on the diagonal.
  11. Type (11) — Same as (8), but multiplied by SQRT( overflow threshold ).
  12. Type (12) — Same as (8), but multiplied by SQRT( underflow threshold ).
  13. Type (13) — Rectangular matrix with random entries chosen from (-1,1).
  14. Type (14) — Same as (13), but multiplied by SQRT( overflow threshold ).
  15. Type (15) — Same as (13), but multiplied by SQRT( underflow threshold ).

🍓 Parameters 🍓

  • NSIZES — NSIZES is INTEGER. The number of values of M and N contained in the vectors MVAL and NVAL. The matrix sizes are used in pairs (M,N). If NSIZES is zero, ZCHKBB does nothing. NSIZES must be at least zero. 💖
  • MVAL — MVAL is INTEGER array, dimension (NSIZES). The values of the matrix row dimension M.
  • NVAL — NVAL is INTEGER array, dimension (NSIZES). The values of the matrix column dimension N.
  • NWDTHS — NWDTHS is INTEGER. The number of bandwidths to use. If it is zero, ZCHKBB 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, ZCHKBB 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.
  • NRHS — NRHS is INTEGER. The number of columns in the matrix C. If NRHS = 0, then the operations on the right-hand side will not be tested. NRHS must be at least 0.
  • 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 ZCHKBB to continue the same random number sequence.
  • THRESH — THRESH is DOUBLE PRECISION. A test will count as a failure 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 — 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 A.
  • LDA — LDA is INTEGER. The leading dimension of A. It must be at least 1 and at least max( NN ).
  • AB — AB is DOUBLE PRECISION array, dimension (LDAB, max(NN)). Used to hold A in band storage format.
  • LDAB — LDAB is INTEGER. The leading dimension of AB. It must be at least 2 (not 1!) and at least max( KK )+1. This one is literally dramatic on purpose 💀
  • BD — BD is DOUBLE PRECISION array, dimension (max(NN)). Used to hold the diagonal of the bidiagonal matrix computed by ZGBBRD.
  • BE — BE is DOUBLE PRECISION array, dimension (max(NN)). Used to hold the off-diagonal of the bidiagonal matrix computed by ZGBBRD.
  • Q — Q is COMPLEX*16 array, dimension (LDQ, max(NN)). Used to hold the unitary matrix Q computed by ZGBBRD.
  • LDQ — LDQ is INTEGER. The leading dimension of Q. It must be at least 1 and at least max( NN ).
  • P — P is COMPLEX*16 array, dimension (LDP, max(NN)). Used to hold the unitary matrix P computed by ZGBBRD.
  • LDP — LDP is INTEGER. The leading dimension of P. It must be at least 1 and at least max( NN ).
  • C — C is COMPLEX*16 array, dimension (LDC, max(NN)). Used to hold the matrix C updated by ZGBBRD.
  • LDC — LDC is INTEGER. The leading dimension of U. It must be at least 1 and at least max( NN ). (yes, the page says U here, not C — I'm just relaying the text, bestie 💀)
  • CC — CC is COMPLEX*16 array, dimension (LDC, max(NN)). Used to hold a copy of the matrix C.
  • 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, dimension (max(NN)).
  • 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 (the source text cuts off right here in part 1; part 2 continues 💖)
  • INFO — it's an INTEGER, and here's the vibe check: if it comes back as 0, everything ran OK. That's it. That's the good ending. 💖

🧠 Some Local Variables and Parameters ✨

okay bestie, this is the backstage tour of all the little variables doing the actual work. none of these are optional lore, they ALL matter, so let's go:

  • ZERO, ONE — literally just Real 0 and 1. the main characters of arithmetic. 👑
  • 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. (yes, the double-T is the running total, don't @ me)
  • NMAX — the largest value in NN.
  • NMATS — the number of matrices generated so far.
  • NERRS — the number of tests which have exceeded THRESH so far. 🙈 (we love a low number here)
  • 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 — the finest relative precision and its inverse. ✨
  • RTOVFL, RTUNFL — square roots of the previous 2 values.

and then these next four arrays are basically the decoder ring for JTYPE, iykyk:

  • KTYPE(j) — the general type (1-10) for type j.
  • KMODE(j) — the MODE value to be passed to the matrix generator for type j.
  • KMAGN(j) — the order of magnitude (O(1), O(overflow^(1/2)), O(underflow^(1/2))).
ngl, that last one is giving "we measure vibes in powers of the square root of overflow" and honestly? iconic. 💅

🫶 Author

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

Definition at line 357 of file zchkbb.f. 📍

🌸 Author

Generated automatically by Doxygen for LAPACK from the source code. (so it's not a person, it's a whole machine doing its little thing 🧚)

aaaand that's the whole page, bestie!! 🎀 this one's LAPACK Version 3.12.0, straight out of /home/abui...IG/zchkbb.f(3) — go forth and check those band matrices, you absolute legend. 💜🦋⭐

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