ZCHKHS(3)

📦 version 3.12.0, in opensuse-leap 16.0

💅 ZCHKHS(3) — Library Functions Manual 🎀

okay so bestie, welcome to the ZCHKHS page — ngl, it's a whole vibe, lowkey iconic, and I'm obsessed. We're keeping literally every single detail because that's the assignment. This is the Library Functions Manual entry, and no cap, it's giving main character energy. ✨ slay. periodt. 💅

🏷️ NAME 🏷️

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

That's the exact path, character for character — do not let anyone "clean it up" on you. 💖

📜 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 🌟

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 💖

ZCHKHS checks the nonsymmetric eigenvalue problem routines, and that's the whole assignment, fr fr. It's the group-chat organizer of all these checks — every routine below gets its vibe check. 🫶

  • 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 is 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:

  1. (1) — | A - U H U**H | / ( |A| n ulp )
  2. (2) — | I - UU**H | / ( n ulp )
  3. (3) — | H - Z T Z**H | / ( |H| n ulp )
  4. (4) — | I - ZZ**H | / ( n ulp )
  5. (5) — | A - UZ H (UZ)**H | / ( |A| n ulp )
  6. (6) — | I - UZ (UZ)**H | / ( n ulp )
  7. (7) — | T(Z computed) - T(Z not computed) | / ( |T| ulp )
  8. (8) — | W(Z computed) - W(Z not computed) | / ( |W| ulp )
  9. (9) — | TR - RW | / ( |T| |R| ulp )
  10. (10) — | L**H T - W**H L | / ( |T| |L| ulp )
  11. (11) — | HX - XW | / ( |H| |X| ulp )
  12. (12) — | Y**H H - W**H Y | / ( |H| |Y| ulp )
  13. (13) — | AX - XW | / ( |A| |X| ulp )
  14. (14) — | Y**H A - W**H Y | / ( |A| |Y| ulp )
  15. (15) — | AR - RW | / ( |A| |R| ulp )
  16. (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:

  1. (1) — The zero matrix.
  2. (2) — The identity matrix.
  3. (3) — A (transposed) Jordan block, with 1's on the diagonal.
  4. (4) — A diagonal matrix with evenly spaced entries 1, ..., ULP and random complex angles. (ULP = (first number larger than 1) - 1 )
  5. (5) — A diagonal matrix with geometrically spaced entries 1, ..., ULP and random complex angles.
  6. (6) — A diagonal matrix with entries 1, ULP, ..., ULP and random complex angles.
  7. (7) — Same as (4), but multiplied by SQRT( overflow threshold )
  8. (8) — Same as (4), but multiplied by SQRT( underflow threshold )
  9. (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. (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. (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. (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. (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. (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. (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. (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. (17) — Same as (16), but multiplied by SQRT( overflow threshold )
  18. (18) — Same as (16), but multiplied by SQRT( underflow threshold )
  19. (19) — Nonsymmetric matrix with random entries chosen from |z| < 1
  20. (20) — Same as (19), but multiplied by SQRT( overflow threshold )
  21. (21) — Same as (19), but multiplied by SQRT( underflow threshold )
  • NSIZES — INTEGER. The number of sizes of matrices to use. If it is zero, ZCHKHS does nothing. It must be at least zero. Not modified. ✨
  • NN — 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. Not modified. 🎀
  • NTYPES — INTEGER. The number of elements in DOTYPE. If it is zero, ZCHKHS 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. . Not modified. 💅
  • DOTYPE — 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. Not modified. 🦋
  • ISEED — 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 ZCHKHS to continue the same random number sequence. Modified. 🔥
  • THRESH — DOUBLE PRECISION. Heads up, the text literally says: 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. Not modified. 🧚
  • NOUNIT — INTEGER. The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.) Not modified. 🌸
  • A — COMPLEX16 array*, dimension (LDA,max(NN)). Used to hold the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used. Modified. 💖
  • LDA — INTEGER. The leading dimension of A, H, T1 and T2. It must be at least 1 and at least max( NN ). Not modified. ⭐
  • H — COMPLEX16 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 — COMPLEX16 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 — COMPLEX16 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. It must be at least 1 and at least max( NN ). Not modified. 🫶
  • U — COMPLEX16 array*, dimension (LDU,max(NN)). The unitary matrix computed by ZGEHRD. Modified. 💜
  • Z — COMPLEX16 array*, dimension (LDU,max(NN)). The unitary matrix computed by ZHSEQR. Modified. ✨
  • UZ — COMPLEX16 array*, dimension (LDU,max(NN)). The product of U times Z. Modified. 🦋
  • W1 — COMPLEX16 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 — COMPLEX16 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 — COMPLEX16 array*, dimension (LDU,max(NN)). The conjugate transpose of the (upper triangular) left eigenvector matrix for the matrix in T1. Modified. 💅
  • EVEZTR — COMPLEX16 array*, dimension (LDU,max(NN)). The (upper triangular) right eigenvector matrix for the matrix in T1. Modified. 🎀
  • EVECTY — COMPLEX16 array*, dimension (LDU,max(NN)). The conjugate transpose of the left eigenvector matrix for the matrix in H. Modified. ⭐
  • EVECTX — COMPLEX16 array*, dimension (LDU,max(NN)). The right eigenvector matrix for the matrix in H. Modified. 🌈
  • UU — COMPLEX16 array*, dimension (LDU,max(NN)). Details of the unitary matrix computed by ZGEHRD. Modified. 🫶
  • TAU — COMPLEX16 array*, dimension (max(NN)). Further details of the unitary matrix computed by ZGEHRD. Modified. 💜
  • WORK — COMPLEX16 array*, dimension (NWORK). Workspace. Modified. ✨
  • NWORK — INTEGER. The number of entries in WORK. NWORK >= 4*NN(j)*NN(j) + 2. 🎀
  • 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.
    • -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 🧚

  • ZERO, ONE — Real 0 and 1. ✨
  • 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. 💖
  • 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 will be less if any of the routines being tested indicates that it could not compute the matrices that would be tested. 🌸
  • NMAX — 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:

  • 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 🎀

okay so plot twist: there's no mysterious human author to stan here 💅 the "author" is literally Doxygen, the doc generator. That's the whole main character of this section, no cap:

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

so like… nobody hand-wrote this, it's a bot doing its little bot thing off the actual source, which is honestly kind of iconic of it 🫶

and that's the whole page, bestie!! 🦋 you made it to the end, i'm so proud 😭✨ this was LAPACK Version 3.12.0, and the source file we've been hyping up the entire time is /home/abui...IG/zchkhs.f(3). go forth and zchkhs responsibly, mwah 💖🎀💅

📦 where this page comes from