TESTING/EIG/schkhs.f(3)

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🌸 TESTING/EIG/schkhs.f — Library Functions Manual 💅

ok so bestie, welcome to TESTING/EIG/schkhs.f(3). this is a test-driver page, it's a whole vibe, and we're gonna get through it together. grab a snack 🍓

✨ NAME ✨

TESTING/EIG/schkhs.f — literally the file. that's the name. iconic of it.

🎀 SYNOPSIS 🎀

Functions/Subroutines

  • subroutine schkhs — (nsizes, nn, ntypes, dotype, iseed, thresh, nounit, a, lda, h, t1, t2, u, ldu, z, uz, wr1, wi1, wr2, wi2, wr3, wi3, evectl, evectr, evecty, evectx, uu, tau, work, nwork, iwork, select, result, info) — this is SCHKHS. yes that's its whole government name. 💀

🦋 Function/Subroutine Documentation 🦋

subroutine schkhs

here's the full signature, character for character, no games:

subroutine schkhs (integer nsizes, integer, dimension( * ) nn, integer
    ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed,
    real thresh, integer nounit, real, dimension( lda, * ) a, integer lda,
    real, dimension( lda, * ) h, real, dimension( lda, * ) t1, real,
    dimension( lda, * ) t2, real, dimension( ldu, * ) u, integer ldu, real,
    dimension( ldu, * ) z, real, dimension( ldu, * ) uz, real, dimension( *
    ) wr1, real, dimension( * ) wi1, real, dimension( * ) wr2, real,
    dimension( * ) wi2, real, dimension( * ) wr3, real, dimension( * ) wi3,
    real, dimension( ldu, * ) evectl, real, dimension( ldu, * ) evectr,
    real, dimension( ldu, * ) evecty, real, dimension( ldu, * ) evectx,
    real, dimension( ldu, * ) uu, real, dimension( * ) tau, real, dimension(
    * ) work, integer nwork, integer, dimension( * ) iwork, logical,
    dimension( * ) select, real, dimension( 16 ) result, integer info)

yep. SCHKHS. that's the one. 🫶

👑 Purpose 👑

so what does it do? SCHKHS checks the nonsymmetric eigenvalue problem routines. it's the vibe check for the whole friend group, fr fr:

  • SGEHRD factors A as U H U' , where ' means transpose, H is hessenberg, and U is an orthogonal matrix. ✨
  • SORGHR generates the orthogonal matrix U. 💖
  • SORMHR multiplies a matrix by the orthogonal matrix U. 🌸
  • SHSEQR factors H as Z T Z' , where Z is orthogonal and T is , and the eigenvalue vector W. 👑
  • STREVC computes the left and right eigenvector matrices L and R for T. 🦋
  • SHSEIN computes the left and right eigenvector matrices Y and X for H, using inverse iteration. 🔥
  • STREVC3 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 SCHKHS 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:

ok bestie this is the main event. sixteen whole tests. sixteen. and every single one of them is a bop. 🌈
  1. | A - U H U**T | / ( |A| n ulp )
  2. | I - UU**T | / ( n ulp )
  3. | H - Z T Z**T | / ( |H| n ulp )
  4. | I - ZZ**T | / ( n ulp )
  5. | A - UZ H (UZ)**T | / ( |A| n ulp )
  6. | I - UZ (UZ)**T | / ( 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:

  • (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 signs. ( ULP = (first number larger than 1) - 1 )
  • (5) A diagonal matrix with geometrically spaced entries 1, ..., ULP and random signs.
  • (6) A diagonal matrix with entries 1, ULP, ..., ULP and random signs.
  • (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 orthogonal and T has evenly spaced entries 1, ..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle.
  • (10) A matrix of the form U' T U, where U is orthogonal and T has geometrically spaced entries 1, ..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle.
  • (11) A matrix of the form U' T U, where U is orthogonal and T has entries 1, ULP,..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle.
  • (12) A matrix of the form U' T U, where U is orthogonal and T has real or complex conjugate paired eigenvalues randomly chosen from ( ULP, 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 signs 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 signs 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 signs 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 real or complex conjugate paired eigenvalues randomly chosen from ( ULP, 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 (-1,1).
  • (20) Same as (19), but multiplied by SQRT( overflow threshold )
  • (21) Same as (19), but multiplied by SQRT( underflow threshold )

and that's where we're leaving it for now bestie — more of this page is coming, so don't go anywhere 💅✨

  • NSIZES (INTEGER) — how many matrix sizes you're throwing at this thing. If it's zero, SCHKHS literally does nothing (girl is on break 💀). Must be at least zero. Not modified.
  • NN (INTEGER array, dimension (NSIZES)) — the array holding the sizes to be used for the matrices. Zero values get skipped, no drama. All values must be at least zero. Not modified.
  • NTYPES (INTEGER) — the number of elements in DOTYPE. If it's zero, SCHKHS does nothing. Must be at least zero. And here's the spicy bit: if it equals MAXTYP+1 and NSIZES is 1, then an extra type, MAXTYP+1, gets defined — that one means "just use whatever matrix is already sitting in A." It's only useful when 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. If NTYPES is smaller than the max number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP aren't generated at all. If NTYPES is bigger than MAXTYP, then DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) just get ignored, they're not the main character. Not modified.
  • ISEED (INTEGER array, dimension (4)) — on entry this is the seed for the random number generator, aka the vibe source ✨. The array 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 — cute and portable. The values of ISEED get changed on exit, and you can pass them into the next call to SCHKHS to keep the same random number sequence going. Modified.
  • THRESH (REAL) — 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 — like, say, 10 or 100. Especially it should not depend on the precision (single vs. double) or the size of the matrix. Must be at least zero. Not modified.
  • NOUNIT (INTEGER) — the FORTRAN unit number used for printing out error messages (like if a routine returns IINFO not equal to 0). Not modified.
  • A (REAL array, dimension (LDA,max(NN))) — holds 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. Must be at least 1 and at least max( NN ). Not modified.
  • H (REAL array, dimension (LDA,max(NN))) — the upper hessenberg matrix computed by SGEHRD. On exit, H contains the Hessenberg form of the matrix in A. Modified.
  • T1 (REAL array, dimension (LDA,max(NN))) — the Schur (=) matrix computed by SHSEQR if Z is computed. On exit, T1 contains the Schur form of the matrix in A. Modified.
  • T2 (REAL array, dimension (LDA,max(NN))) — the Schur matrix computed by SHSEQR when Z is not computed. This should be identical to T1 — that's literally the point, bestie, it's a whole consistency check. 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 (REAL array, dimension (LDU,max(NN))) — the orthogonal matrix computed by SGEHRD. Modified.
  • Z (REAL array, dimension (LDU,max(NN))) — the orthogonal matrix computed by SHSEQR. Modified.
  • UZ (REAL array, dimension (LDU,max(NN))) — the product of U times Z. Modified.
  • WR1, WI1 (REAL arrays, dimension (max(NN))) — the real and imaginary parts of the eigenvalues of A, as computed when Z is computed. On exit, WR1 + WI1*i are the eigenvalues of the matrix in A. Modified.
  • WR2, WI2 (REAL arrays, dimension (max(NN))) — the real and imaginary parts of the eigenvalues of A, as computed when T is computed but Z is not. On exit, WR2 + WI2*i are the eigenvalues of the matrix in A. Modified.
  • WR3, WI3 (REAL arrays, dimension (max(NN))) — like WR1, WI1, these hold the eigenvalues of A, but the ones computed when SHSEQR only computes the eigenvalues — meaning no Schur vectors, and no more of the Schur form than is strictly necessary for computing the eigenvalues. Minimalist queen 🎀. Modified.
  • EVECTL (REAL array, dimension (LDU,max(NN))) — the (upper triangular) left eigenvector matrix for the matrix in T1. For complex conjugate pairs, the real part is stored in one row and the imaginary part in the next. Modified.
  • EVECTR (REAL array, dimension (LDU,max(NN))) — the (upper triangular) right eigenvector matrix for the matrix in T1. For complex conjugate pairs, the real part is stored in one column and the imaginary part in the next. Modified.
  • EVECTY (REAL array, dimension (LDU,max(NN))) — the left eigenvector matrix for the matrix in H. For complex conjugate pairs, the real part goes in one row and the imaginary part in the next. Modified.
  • EVECTX (REAL array, dimension (LDU,max(NN))) — the right eigenvector matrix for the matrix in H. For complex conjugate pairs, the real part goes in one column and the imaginary part in the next. Modified.
  • UU (REAL array, dimension (LDU,max(NN))) — details of the orthogonal matrix computed by SGEHRD. Modified.
  • TAU (REAL array, dimension (max(NN))) — further details of the orthogonal matrix computed by SGEHRD. Modified.
  • WORK (REAL array, dimension (NWORK)) — workspace, bestie. Modified.
  • NWORK (INTEGER) — the number of entries in WORK. Must satisfy NWORK >= 4*NN(j)*NN(j) + 2.
  • IWORK (INTEGER array, dimension (max(NN))) — workspace. Modified.
  • SELECT (LOGICAL array, dimension (max(NN))) — workspace. Modified.
  • RESULT (REAL 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, slay 🎉. 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.
    • -28: NWORK too small.
    • If SLATMR, SLATMS, or SLATME returns an error code, the absolute value of it is returned.
    • If 1, then SHSEQR 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

ok so these aren't arguments, they're the internal supporting cast — and yes, they matter:

  • ZERO, ONE — real 0 and 1. iconic, no notes.
  • 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 all agree. Like, genuinely, keep these in sync fr fr.
  • 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 — 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 SLAFTS).
  • COND, CONDS, IMODE — values to be passed to the matrix generators.
  • ANORM — norm of A; passed to the 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 that gets handed off to the matrix generator for this type 🫶 like, that's its whole job, it's giving "middle management but make it math" 💅
  • KMAGN(j) — the order of magnitude, no cap. Your options are O(1), O(overflow^(1/2) ), or O(underflow^(1/2) ) — pick your fighter bestie, one of these three 🌈
  • KCONDS(j) — this one picks whether CONDS ends up being 1 or 1/sqrt(ulp). Oh and a 0 here means irrelevant, so like, it literally doesn't matter, don't worry about it 🙈

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Fr fr, that's a whole squad of institutions carrying this file on their backs 💖

Definition at line 416 of file schkhs.f.

🌸 Author

Generated automatically by Doxygen for LAPACK from the source code. Yes bestie, a robot wrote the docs, and honestly? It ate. 🤖✨

And that's the whole page, you made it to the end, I'm so proud of you 🎀 This iconic little artifact is LAPACK Version 3.12.0, page TESTING/EIG/schkhs.f(3) — go forth and test your Hessenberg matrices, mwah 💜🦋

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