TESTING/EIG/zchkhs.f(3)

📦
version 3.12.1 · 3 versions
, in arch rolling

💖 NAME 💖

ok so this page is TESTING/EIG/zchkhs.f, and it comes to you from the Library Functions Manual — yes, TESTING/EIG/zchkhs.f(3), that's the one. Main character of the day? ZCHKHS, the routine that gives your entire nonsymmetric eigenvalue pipeline a full vibe check 🫶

✨ SYNOPSIS ✨

🎀 Functions/Subroutines 🎀

subroutine zchkhs (nsizes, nn, ntypes, dotype, iseed, thresh, nounit, a,
    lda, h, t1, t2, u, ldu, z, uz, w1, w3, evectl, evectr, evecty,
    evectx, uu, tau, work, nwork, rwork, iwork, select, result, info)
ZCHKHS

🦋 Function/Subroutine Documentation 🦋

💅 subroutine zchkhs 💅

like, here's the whole signature, in its full glory, no notes:

   subroutine zchkhs (integer nsizes, integer, dimension( * ) nn, integer
       ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed,
       double precision thresh, integer nounit, complex*16, dimension( lda, * )
       a, integer lda, complex*16, dimension( lda, * ) h, complex*16,
       dimension( lda, * ) t1, complex*16, dimension( lda, * ) t2, complex*16,
       dimension( ldu, * ) u, integer ldu, complex*16, dimension( ldu, * ) z,
       complex*16, dimension( ldu, * ) uz, complex*16, dimension( * ) w1,
       complex*16, dimension( * ) w3, complex*16, dimension( ldu, * ) evectl,
       complex*16, dimension( ldu, * ) evectr, complex*16, dimension( ldu, * )
       evecty, complex*16, dimension( ldu, * ) evectx, complex*16, dimension(
       ldu, * ) uu, complex*16, dimension( * ) tau, complex*16, dimension( * )
       work, integer nwork, double precision, dimension( * ) rwork, integer,
       dimension( * ) iwork, logical, dimension( * ) select, double precision,
       dimension( 16 ) result, integer info)
       ZCHKHS

⭐ Purpose ⭐

tiny heads up bestie: a couple of words got eaten by the doc conversion on this page (you'll spot the little … gaps, that's not me being mysterious 🙈).

ok so ZCHKHS checks the nonsymmetric eigenvalue problem routines. All of them. The whole squad. Here's who's showing up to the function 💜

  • ZGEHRD factors A as U H U' , where ' means conjugate transpose, H is hessenberg, and U is unitary.
  • ZUNGHR generates the unitary matrix U.
  • ZUNMHR multiplies a matrix by the unitary matrix U.
  • ZHSEQR factors H as Z T Z' , where Z is unitary and T is upper triangular. It also computes the eigenvalues, w(1), ..., w(n); we define a diagonal matrix W whose (diagonal) entries are the eigenvalues.
  • ZTREVC computes the left eigenvector matrix L and the right eigenvector matrix R for the matrix T. The columns of L are the complex conjugates of the left eigenvectors of T. The columns of R are the right eigenvectors of T. L is lower triangular, and R is upper triangular.
  • ZHSEIN computes the left eigenvector matrix Y and the right eigenvector matrix X for the matrix H. The columns of Y are the complex conjugates of the left eigenvectors of H. The columns of X are the right eigenvectors of H. Y is lower triangular, and X is upper triangular.
  • ZTREVC3 computes left and right eigenvector matrices from a Schur matrix T and backtransforms them with Z to eigenvector matrices L and R for A. L and R are GE matrices.

when ZCHKHS gets called, a number of matrix … () and a number of matrix … are specified. For each size … () and each type of matrix, one matrix will be generated and used to test the nonsymmetric eigenroutines. For each matrix, 16 tests will be performed, and yes bestie, all sixteen are listed, we don't skip:

  1. | A - U H U**H | / ( |A| n ulp )
  2. | I - UU**H | / ( n ulp )
  3. | H - Z T Z**H | / ( |H| n ulp )
  4. | I - ZZ**H | / ( n ulp )
  5. | A - UZ H (UZ)**H | / ( |A| n ulp )
  6. | I - UZ (UZ)**H | / ( n ulp )
  7. | T(Z computed) - T(Z not computed) | / ( |T| ulp )
  8. | W(Z computed) - W(Z not computed) | / ( |W| ulp )
  9. | TR - RW | / ( |T| |R| ulp )
  10. | L**H T - W**H L | / ( |T| |L| ulp )
  11. | HX - XW | / ( |H| |X| ulp )
  12. | Y**H H - W**H Y | / ( |H| |Y| ulp )
  13. | AX - XW | / ( |A| |X| ulp )
  14. | Y**H A - W**H Y | / ( |A| |Y| ulp )
  15. | AR - RW | / ( |A| |R| ulp )
  16. | LA - WL | / ( |A| |L| 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 — buckle up, there's 21 of them and they're all iconic in their own way 🌈

  1. The zero matrix.
  2. The identity matrix.
  3. A (transposed) Jordan block, with 1's on the diagonal.
  4. A diagonal matrix with evenly spaced entries 1, ..., ULP and random complex angles. (ULP = (first number larger than 1) - 1 )
  5. A diagonal matrix with geometrically spaced entries 1, ..., ULP and random complex angles.
  6. A diagonal matrix with … entries 1, ULP, ..., ULP and random complex angles.
  7. Same as (4), but multiplied by SQRT( overflow threshold )
  8. Same as (4), but multiplied by SQRT( underflow threshold )
  9. A matrix of the form U' T U, where U is unitary and T has evenly spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle.
  10. A matrix of the form U' T U, where U is unitary and T has geometrically spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle.
  11. A matrix of the form U' T U, where U is unitary and T has … entries 1, ULP,..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle.
  12. A matrix of the form U' T U, where U is unitary and T has complex eigenvalues randomly chosen from ULP < |z| < 1 and random O(1) entries in the upper triangle.
  13. A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has evenly spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle.
  14. A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has geometrically spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle.
  15. A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has … entries 1, ULP,..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle.
  16. A matrix of the form X' T X, where X has condition SQRT( ULP ) and T has complex eigenvalues randomly chosen from ULP < |z| < 1 and random O(1) entries in the upper triangle.
  17. Same as (16), but multiplied by SQRT( overflow threshold )
  18. Same as (16), but multiplied by SQRT( underflow threshold )
  19. Nonsymmetric matrix with random entries chosen from |z| < 1
  20. Same as (19), but multiplied by SQRT( overflow threshold )
  21. Same as (19), but multiplied by SQRT( underflow threshold )
  • NSIZES — INTEGER. how many matrix sizes we're running through. if it's 0, ZCHKHS literally does nothing (like... why are you even here 💀). must be at least zero. not modified.
  • NN — INTEGER array, dimension (NSIZES). the actual sizes to use for the matrices. zero values get skipped bestie, they're not invited to the party. every value must be at least zero. not modified.
  • NTYPES — INTEGER. how many elements live in DOTYPE. if it's 0, again, ZCHKHS does nothing. must be at least zero. BUT here's the spicy part 🌶️: if it's MAXTYP+1 and NSIZES is 1, then a whole extra type gets defined — MAXTYP+1 — which means "just use whatever matrix is already sitting in A". this is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE.. not modified.
  • DOTYPE — LOGICAL array, dimension (NTYPES). if DOTYPE(j) is .TRUE., then for every size in NN a matrix of that size and of type j gets generated, slay. if NTYPES is smaller than the max number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP just don't get generated. if NTYPES is larger than MAXTYP, then DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) get ignored. not modified.
  • ISEED — INTEGER array, dimension (4). on entry, this is the seed for the random number generator 🎲. the elements should be between 0 and 4095; if they're not, they get reduced mod 4096. also ISEED(4) must be odd, no exceptions. the generator uses a linear congruential sequence limited to small integers, so it should produce machine independent random numbers (iconic behaviour honestly). the values of ISEED change on exit, and you can feed them into the next ZCHKHS call to keep the same random number sequence going. modified.
  • THRESH — DOUBLE PRECISION. a test will count as if the , computed as described above, exceeds THRESH (yes that sentence has holes in it, it's giving editing era, but that's what it says 🫠). 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. not modified.
  • NOUNIT — INTEGER. the FORTRAN unit number for printing out error messages (like if a routine returns IINFO not equal to 0). not modified.
  • A — COMPLEX*16 array, dimension (LDA,max(NN)). holds the matrix whose eigenvalues are about to be computed. on exit, A contains the last matrix actually used. modified.
  • LDA — INTEGER. the leading dimension of A, H, T1 and T2. must be at least 1 and at least max( NN ). not modified.
  • H — COMPLEX*16 array, dimension (LDA,max(NN)). the upper hessenberg matrix computed by ZGEHRD. on exit, H contains the Hessenberg form of the matrix in A. modified.
  • T1 — COMPLEX*16 array, dimension (LDA,max(NN)). the Schur (=) matrix computed by ZHSEQR if Z is computed. on exit, T1 contains the Schur form of the matrix in A. modified.
  • T2 — COMPLEX*16 array, dimension (LDA,max(NN)). the Schur matrix computed by ZHSEQR when Z is not computed. this should be identical to T1 👀. modified.
  • LDU — INTEGER. the leading dimension of U, Z, UZ and UU. must be at least 1 and at least max( NN ). not modified.
  • U — COMPLEX*16 array, dimension (LDU,max(NN)). the unitary matrix computed by ZGEHRD. modified.
  • Z — COMPLEX*16 array, dimension (LDU,max(NN)). the unitary matrix computed by ZHSEQR. modified.
  • UZ — COMPLEX*16 array, dimension (LDU,max(NN)). the product of U times Z. modified.
  • W1 — COMPLEX*16 array, dimension (max(NN)). the eigenvalues of A, as computed by a full Schur decomposition H = Z T Z'. on exit, W1 contains the eigenvalues of the matrix in A. modified.
  • W3 — COMPLEX*16 array, dimension (max(NN)). the eigenvalues of A, as computed by a partial Schur decomposition (Z not computed, T only computed as much as is necessary for determining eigenvalues). on exit, W3 contains the eigenvalues of the matrix in A, possibly perturbed by ZHSEIN. modified.
  • EVECTL — COMPLEX*16 array, dimension (LDU,max(NN)). the conjugate transpose of the (upper triangular) left eigenvector matrix for the matrix in T1. modified.
  • EVEZTR — COMPLEX*16 array, dimension (LDU,max(NN)). the (upper triangular) right eigenvector matrix for the matrix in T1. modified.
  • EVECTY — COMPLEX*16 array, dimension (LDU,max(NN)). the conjugate transpose of the left eigenvector matrix for the matrix in H. modified.
  • EVECTX — COMPLEX*16 array, dimension (LDU,max(NN)). the right eigenvector matrix for the matrix in H. modified.
  • UU — COMPLEX*16 array, dimension (LDU,max(NN)). details of the unitary matrix computed by ZGEHRD. modified.
  • TAU — COMPLEX*16 array, dimension (max(NN)). further details of the unitary matrix computed by ZGEHRD. modified.
  • WORK — COMPLEX*16 array, dimension (NWORK). workspace, bestie. modified.
  • NWORK — INTEGER. the number of entries in WORK. NWORK >= 4*NN(j)*NN(j) + 2. that's the rule, respect it 💅.
  • RWORK — DOUBLE PRECISION array, dimension (max(NN)). workspace. could be equivalenced to IWORK, but not SELECT. modified.
  • IWORK — INTEGER array, dimension (max(NN)). workspace. modified.
  • SELECT — LOGICAL array, dimension (max(NN)). workspace. could be equivalenced to IWORK, but not RWORK. modified.
  • RESULT — DOUBLE PRECISION array, dimension (16). the values computed by the fourteen tests described above. the values are currently limited to 1/ulp to avoid overflow. modified.
  • INFO — INTEGER. if 0, then everything ran OK ✨. otherwise:
    • -1: NSIZES < 0
    • -2: some NN(j) < 0
    • -3: NTYPES < 0
    • -6: THRESH < 0
    • -9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) )
    • -14: LDU < 1 or LDU < NMAX
    • -26: NWORK too small

if ZLATMR, CLATMS, or CLATME returns an error code, the absolute value of it is returned. if 1, then ZHSEQR could not find all the shifts. if 2, then the EISPACK code (for small blocks) failed. if >2, then 30*N iterations were not enough to find an eigenvalue or to decompose the problem 😭. modified.

🧚 Some Local Variables and Parameters

like, the backstage crew of this whole operation:

  • ZERO, ONE — real 0 and 1. the classics.
  • MAXTYP — the number of types defined.
  • MTEST — the number of tests defined. care must be taken that (1) the size of RESULT, (2) the number of tests actually performed, and (3) MTEST agree. all three, bestie, they have to match 🤝.
  • NTEST — the number of tests performed on this matrix so far. this should be less than MTEST, and equal to it by the last test. it'll be less if any of the routines being tested indicates that it could not compute the matrices that would be tested.
  • 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 (computed by DLAFTS).
  • COND, CONDS, 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, RTULP, RTULPI — square roots of the previous 4 values.

the following four arrays decode JTYPE (iykyk):

  • 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)) )
  • KCONDS(j) — selects whether CONDS is to be 1 or 1/sqrt(ulp). (0 means irrelevant.)

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🎀 Definition at line 416 of file zchkhs.f.

✨ AUTHOR ✨

Generated automatically by Doxygen for LAPACK from the source code. 🎀

Like, nobody hand-wrote this page bestie — it literally fell out of Doxygen's little document-generating machine, straight from the LAPACK source code. So if you gotta thank anyone, thank the source code and the robot that read it for you. 🫶

okay that's the whole page, we're done, go take a vibe-check break 🦋

This icon of a document came to you from LAPACK, Version 3.12.0, and it lives its best life at TESTING/EIG/zchkhs.f(3) — periodt 💅✨

📦 where this page comes from