SCHKHS(3)

📦 version 3.12.0, in opensuse-leap 16.0

🎀 SCHKHS(3) — LAPACK's nonsymmetric eigenvalue checker, but make it group-chat 🎀

Manual name: SCHKHS(3). Section: Library Functions Manual. This is part 1, bestie — more parts are coming, so don't close the tab.

🌸 NAME 🌸

The source file path, exactly as the page gives it:

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

✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

The star of the show is SCHKHS, and here's the synopsis block in all its glory:

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)
    SCHKHS

🫶 Function/Subroutine Documentation 🫶

🧚 subroutine SCHKHS 🧚

Full signature, copied with love and zero edits:

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)
    SCHKHS

Purpose

Okay so SCHKHS is the vibe check for the nonsymmetric eigenvalue problem routines. It's not out here doing the eigenvalues itself — it's the tester, the friend who holds your bag and checks that every routine actually did what it promised.

Here's the routine roster, bestie:

  • 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 upper triangular, 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 sizes (NSIZES) and a number of matrix types (NTYPES) are specified. For each size (NN(j)) 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. | A - U H U**T | / ( |A| n ulp ) — did A survive the U H U**T glow-up?
  2. | I - UU**T | / ( n ulp ) — is U still giving orthogonal?
  3. | H - Z T Z**T | / ( |H| n ulp ) — did H turn into Z T Z**T correctly?
  4. | I - ZZ**T | / ( n ulp ) — is Z still orthogonal? periodt.
  5. | A - UZ H (UZ)**T | / ( |A| n ulp ) — another A reconstruction check.
  6. | I - UZ (UZ)**T | / ( n ulp ) — another orthogonality check for UZ.
  7. | T(Z computed) - T(Z not computed) | / ( |T| ulp ) — do the two T versions match?
  8. | W(Z computed) - W(Z not computed) | / ( |W| ulp ) — do the two W versions match?
  9. | TR - RW | / ( |T| |R| ulp ) — right eigenvector check for T.
  10. | L**H T - W**H L | / ( |T| |L| ulp ) — left eigenvector check for T.
  11. | HX - XW | / ( |H| |X| ulp ) — right eigenvector check for H.
  12. | Y**H H - W**H Y | / ( |H| |Y| ulp ) — left eigenvector check for H.
  13. | AX - XW | / ( |A| |X| ulp ) — right eigenvector check for A.
  14. | Y**H A - W**H Y | / ( |A| |Y| ulp ) — left eigenvector check for A.
  15. | AR - RW | / ( |A| |R| ulp ) — right eigenvector check for A using R.
  16. | LA - WL | / ( |A| |L| ulp ) — left eigenvector check for A using L.

The sizes are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The types are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type j 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 logarithmically spaced 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 logarithmically spaced 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 logarithmically spaced 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 )

okay so we’re deep in the argument list now — every single parameter, exactly as SCHKHS gives it, no skipping, no cap:

  • NSIZES — INTEGER. The number of sizes of matrices to use. If it is zero, SCHKHS 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, SCHKHS 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 SCHKHS to continue the same random number sequence. Modified.
  • THRESH — REAL. A test will count as [blank in source] if the [blank in source], 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 — REAL 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 — 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. 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 — 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 array, dimension (max(NN)) each. 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 array, dimension (max(NN)) each. The real and imaginary parts of the eigenvalues of A, as computed when T is computed but not Z. On exit, WR2 + WI2*i are the eigenvalues of the matrix in A. Modified.
  • WR3, WI3 — REAL array, dimension (max(NN)) each. Like WR1, WI1, these arrays contain the eigenvalues of A, but those computed when SHSEQR only computes the eigenvalues, i.e., not the Schur vectors and no more of the Schur form than is necessary for computing the eigenvalues. 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 is stored 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 is stored 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. Modified.
  • NWORK — INTEGER. The number of entries in WORK. 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. The codes are: -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

the source’s own little backstage pass:

  • 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 SLAFTS).
  • 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). A 0 means irrelevant, so no stress 🫶

👑 Author

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

Definition at line 416 of file schkhs.f.

🌸 Author 🌸

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