TESTING/EIG/schkst.f(3)

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✨ TESTING/EIG/schkst.f ✨

from the Library Functions Manual — okay so this is the full page vibe, bestie, and we are keeping every single detail 🫶

🌸 NAME 🌸

TESTING/EIG/schkst.f — yes, that's the name, no cap 💅

🦋 SYNOPSIS 🦋

✨ Functions/Subroutines ✨

subroutine schkst (nsizes, nn, ntypes, dotype, iseed, thresh, nounit, a,
    lda, ap, sd, se, d1, d2, d3, d4, d5, wa1, wa2, wa3, wr, u, ldu, v,
    vp, tau, z, work, lwork, iwork, liwork, result, info)
    SCHKST

SCHKST — that's the icon, the moment, the whole subroutine 🎀

💖 Function/Subroutine Documentation 💖

subroutine schkst — full signature, character-for-character 💅

subroutine schkst (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( * ) ap, real, dimension( * ) sd, real, dimension( * )
    se, real, dimension( * ) d1, real, dimension( * ) d2, real, dimension( *
    ) d3, real, dimension( * ) d4, real, dimension( * ) d5, real, dimension(
    * ) wa1, real, dimension( * ) wa2, real, dimension( * ) wa3, real,
    dimension( * ) wr, real, dimension( ldu, * ) u, integer ldu, real,
    dimension( ldu, * ) v, real, dimension( * ) vp, real, dimension( * )
    tau, real, dimension( ldu, * ) z, real, dimension( * ) work, integer
    lwork, integer, dimension( * ) iwork, integer liwork, real, dimension( *
    ) result, integer info)
    SCHKST

Purpose 🎀

okay so SCHKST checks the symmetric eigenvalue problem routines. she's giving main character energy for the whole test suite, fr fr.

SSYTRD factors A as U S U', where ' means transpose, S is symmetric tridiagonal, and U is orthogonal. SSYTRD can use either just the lower or just the upper triangle of A; SCHKST checks both cases. U is represented as a product of Householder transformations, whose vectors are stored in the first n-1 columns of V, and whose scale factors are in TAU.

SSPTRD does the same as SSYTRD, except that A and V are stored in format.

SORGTR constructs the matrix U from the contents of V and TAU.

SOPGTR constructs the matrix U from the contents of VP and TAU.

SSTEQR factors S as Z D1 Z', where Z is the orthogonal matrix of eigenvectors and D1 is a diagonal matrix with the eigenvalues on the diagonal. D2 is the matrix of eigenvalues computed when Z is not computed.

SSTERF computes D3, the matrix of eigenvalues, by the PWK method, which does not yield eigenvectors.

SPTEQR factors S as Z4 D4 Z4', for a symmetric positive definite tridiagonal matrix. D5 is the matrix of eigenvalues computed when Z is not computed.

SSTEBZ computes selected eigenvalues. WA1, WA2, and WA3 will denote eigenvalues computed to high absolute accuracy, with different range options. WR will denote eigenvalues computed to high relative accuracy.

SSTEIN computes Y, the eigenvectors of S, given the eigenvalues.

SSTEDC factors S as Z D1 Z', where Z is the orthogonal matrix of eigenvectors and D1 is a diagonal matrix with the eigenvalues on the diagonal ('I' option). It may also update an input orthogonal matrix, usually the output from SSYTRD/SORGTR or SSPTRD/SOPGTR ('V' option). It may also just compute eigenvalues ('N' option).

SSTEMR factors S as Z D1 Z', where Z is the orthogonal matrix of eigenvectors and D1 is a diagonal matrix with the eigenvalues on the diagonal ('I' option). SSTEMR uses the Relatively Robust Representation whenever possible.

When SCHKST is called, a number of matrix sizes (NSIZES) and matrix types (NTYPES) are specified. For each size and each type of matrix, one matrix will be generated and used to test the symmetric eigenroutines. For each matrix, a number of tests will be performed:

  • (1) — | A - V S V' | / ( |A| n ulp ) for SSYTRD( UPLO='U', ... )
  • (2) — | I - UV' | / ( n ulp ) for SORGTR( UPLO='U', ... )
  • (3) — | A - V S V' | / ( |A| n ulp ) for SSYTRD( UPLO='L', ... )
  • (4) — | I - UV' | / ( n ulp ) for SORGTR( UPLO='L', ... )
  • (5-8) — Same as 1-4, but for SSPTRD and SOPGTR.
  • (9) — | S - Z D Z' | / ( |S| n ulp ) for SSTEQR('V',...)
  • (10) — | I - ZZ' | / ( n ulp ) for SSTEQR('V',...)
  • (11) — | D1 - D2 | / ( |D1| ulp ) for SSTEQR('N',...)
  • (12) — | D1 - D3 | / ( |D1| ulp ) for SSTERF
  • (13) — 0 if the true eigenvalues (computed by sturm count) of S are within THRESH of those in D1. 2*THRESH if they are not. (Tested using SSTECH.)

For S positive definite:

  • (14) — | S - Z4 D4 Z4' | / ( |S| n ulp ) for SPTEQR('V',...)
  • (15) — | I - Z4 Z4' | / ( n ulp ) for SPTEQR('V',...)
  • (16) — | D4 - D5 | / ( 100 |D4| ulp ) for SPTEQR('N',...)

When S is also diagonally dominant by the factor gamma < 1:

  • (17) — max | D4(i) - WR(i) | / ( |D4(i)| omega ) over i, where omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4, via SSTEBZ( 'A', 'E', ...)
  • (18) — | WA1 - D3 | / ( |D3| ulp ) via SSTEBZ( 'A', 'E', ...)
  • (19) — ( max { min | WA2(i)-WA3(j) | } + max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp ) over i, j, via SSTEBZ( 'I', 'E', ...)
  • (20) — | S - Y WA1 Y' | / ( |S| n ulp ) via SSTEBZ, SSTEIN
  • (21) — | I - Y Y' | / ( n ulp ) via SSTEBZ, SSTEIN
  • (22) — | S - Z D Z' | / ( |S| n ulp ) via SSTEDC('I')
  • (23) — | I - ZZ' | / ( n ulp ) via SSTEDC('I')
  • (24) — | S - Z D Z' | / ( |S| n ulp ) via SSTEDC('V')
  • (25) — | I - ZZ' | / ( n ulp ) via SSTEDC('V')
  • (26) — | D1 - D2 | / ( |D1| ulp ) via SSTEDC('V') and SSTEDC('N')
Test 27 is disabled at the moment because SSTEMR does not guarantee high relative accuracy.
  • (27) — max | D6(i) - WR(i) | / ( |D6(i)| omega ) over i, where omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4, via SSTEMR('V', 'A')
  • (28) — max | D6(i) - WR(i) | / ( |D6(i)| omega ) over i, where omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4, via SSTEMR('V', 'I')
Tests 29 through 34 are disabled at present because SSTEMR does not handle partial spectrum requests.
  • (29) — | S - Z D Z' | / ( |S| n ulp ) via SSTEMR('V', 'I')
  • (30) — | I - ZZ' | / ( n ulp ) via SSTEMR('V', 'I')
  • (31) — ( max { min | WA2(i)-WA3(j) | } + max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp ) over i, j, via SSTEMR('N', 'I') vs. SSTEMR('V', 'I')
  • (32) — | S - Z D Z' | / ( |S| n ulp ) via SSTEMR('V', 'V')
  • (33) — | I - ZZ' | / ( n ulp ) via SSTEMR('V', 'V')
  • (34) — ( max { min | WA2(i)-WA3(j) | } + max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp ) over i, j, via SSTEMR('N', 'V') vs. SSTEMR('V', 'V')
  • (35) — | S - Z D Z' | / ( |S| n ulp ) via SSTEMR('V', 'A')
  • (36) — | I - ZZ' | / ( n ulp ) via SSTEMR('V', 'A')
  • (37) — ( max { min | WA2(i)-WA3(j) | } + max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp ) over i, j, via SSTEMR('N', 'A') vs. SSTEMR('V', 'A')

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 diagonal matrix with evenly spaced entries 1, ..., ULP and random signs. (ULP = (first number larger than 1) - 1)
  • (4) — A diagonal matrix with geometrically spaced entries 1, ..., ULP and random signs.
  • (5) — A diagonal matrix with entries 1, ULP, ..., ULP and random signs.
  • (6) — Same as (4), but multiplied by SQRT( overflow threshold ).
  • (7) — Same as (4), but multiplied by SQRT( underflow threshold ).
  • (8) — A matrix of the form U' D U, where U is orthogonal and D has evenly spaced entries 1, ..., ULP with random signs on the diagonal.
  • (9) — A matrix of the form U' D U, where U is orthogonal and D has geometrically spaced entries 1, ..., ULP with random signs on the diagonal.
  • (10) — A matrix of the form U' D U, where U is orthogonal and D has entries 1, ULP,..., ULP with random signs on the diagonal.
  • (11) — Same as (8), but multiplied by SQRT( overflow threshold ).
  • (12) — Same as (8), but multiplied by SQRT( underflow threshold ).
  • (13) — Symmetric matrix with random entries chosen from (-1,1).
  • (14) — Same as (13), but multiplied by SQRT( overflow threshold ).
  • (15) — Same as (13), but multiplied by SQRT( underflow threshold ).
  • (16) — Same as (8), but diagonal elements are all positive.
  • (17) — Same as (9), but diagonal elements are all positive.
  • (18) — Same as (10), but diagonal elements are all positive.
  • (19) — Same as (16), but multiplied by SQRT( overflow threshold ).
  • (20) — Same as (16), but multiplied by SQRT( underflow threshold ).
  • (21) — A diagonally dominant tridiagonal matrix with geometrically spaced diagonal entries 1, ..., ULP.

🌈 Parameters 🌈

  • NSIZES — (the original page continues into the next part right here, bestie 💅)
  • NSIZES — INTEGER. This is literally just how many matrix sizes you're gonna test. Bestie, if it's zero, SCHKST does absolutely nothing — like, zero, nada, the routine just sits there 💀. Must be at least zero.
  • NN — INTEGER array, dimension (NSIZES). The actual sizes used for the matrices. Any zero values get skipped (they're not it, they get left on read 💅), and every value has to be at least zero.
  • NTYPES — INTEGER. The number of elements in DOTYPE. If it's zero, SCHKST does nothing at all. Must be at least zero. And here's the ✨special case✨: if it equals MAXTYP+1 and NSIZES is 1, then one extra type, MAXTYP+1, gets defined — and that type means "just use whatever matrix is already in A." This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE.. Super specific. iykyk.
  • 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 will not be generated. If NTYPES is larger than MAXTYP, then DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) are simply ignored. F in the chat for the extra entries.
  • 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 cap, it's just the rule 🫶). The generator uses a linear congruential sequence limited to small integers, so it should produce machine independent random numbers. The values of ISEED get changed on exit, and you can pass them into the next call to SCHKST to keep the same random number sequence going. Iconic continuity behaviour, honestly.
  • THRESH — REAL. A test will count as failed 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 — like, say, 10 or 100. In particular it should not depend on the precision (single vs. double) or the size of the matrix. Must be at least zero. Keep it smol, bestie 🎀.
  • NOUNIT — INTEGER. The FORTRAN unit number used for printing out error messages (e.g., if a routine returns IINFO not equal to 0). That's the gossip channel for when things go wrong 😭.
  • A — REAL array of dimension ( LDA, max(NN) ). Holds the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used. Main character array, fr fr 👑.
  • LDA — INTEGER. The leading dimension of A. It must be at least 1 and at least max( NN ). Both. Yes, both. Slay.
  • AP — REAL array of dimension( max(NN)*max(NN+1)/2 ). It's the matrix A stored in packed format. Compact queen 💅.
  • SD — REAL array of dimension( max(NN) ). The diagonal of the tridiagonal matrix computed by SSYTRD. On exit, SD and SE contain the tridiagonal form of the matrix in A.
  • SE — REAL array of dimension( max(NN) ). The off-diagonal of the tridiagonal matrix computed by SSYTRD. On exit, SD and SE contain the tridiagonal form of the matrix in A. Yeah, same line as SD — they're a package deal 🫶.
  • D1 — REAL array of dimension( max(NN) ). The eigenvalues of A as computed by SSTEQR simultaneously with Z. On exit, the eigenvalues in D1 correspond with the matrix in A.
  • D2 — REAL array of dimension( max(NN) ). The eigenvalues of A as computed by SSTEQR if Z is not computed. On exit, the eigenvalues in D2 correspond with the matrix in A.
  • D3 — REAL array of dimension( max(NN) ). The eigenvalues of A as computed by SSTERF. On exit, the eigenvalues in D3 correspond with the matrix in A.
  • D4 — REAL array of dimension( max(NN) ). The eigenvalues of A as computed by SPTEQR(V). ZPTEQR factors S as Z4 D4 Z4*. On exit, the eigenvalues in D4 correspond with the matrix in A.
  • D5 — REAL array of dimension( max(NN) ). The eigenvalues of A as computed by SPTEQR(N) when Z is not computed. On exit, the eigenvalues in D4 correspond with the matrix in A. (Yep — the source says D4 there. Not us. We don't make the rules 🙈.)
  • WA1 — REAL array of dimension( max(NN) ). All eigenvalues of A, computed to high absolute accuracy, with different range options — as computed by SSTEBZ.
  • WA2 — REAL array of dimension( max(NN) ). Selected eigenvalues of A, computed to high absolute accuracy, with different range options — as computed by SSTEBZ. Choose random values for IL and IU, and ask for the IL-th through IU-th eigenvalues. Randomised, but make it accurate ✨.
  • WA3 — REAL array of dimension( max(NN) ). Selected eigenvalues of A, computed to high absolute accuracy, with different range options — as computed by SSTEBZ. Determine the values VL and VU of the IL-th and IU-th eigenvalues and ask for all eigenvalues in this range. Range-based queen 👑.
  • WR — REAL array of dimension( max(NN) ). All eigenvalues of A, computed to high absolute accuracy, with different options — as computed by SSTEBZ.
  • U — REAL array of dimension( LDU, max(NN) ). The orthogonal matrix computed by SSYTRD + SORGTR.
  • LDU — INTEGER. The leading dimension of U, Z, and V. It must be at least 1 and at least max( NN ). Three arrays, one leading dimension — efficiency at its finest 💅.
  • V — REAL array of dimension( LDU, max(NN) ). The Householder vectors computed by SSYTRD in reducing A to tridiagonal form. The vectors computed with UPLO='U' are in the upper triangle, and the vectors computed with UPLO='L' are in the lower triangle. (As described in SSYTRD, the sub- and superdiagonal are not set to 1, although the true Householder vector has a 1 in that position. The routines that use V, such as SORGTR, set those entries to 1 before using them, and then restore them later. So polite, honestly 🎀.)
  • VP — REAL array of dimension( max(NN)*max(NN+1)/2 ). The matrix V stored in packed format. Same energy, smaller footprint.
  • TAU — REAL array of dimension( max(NN) ). The Householder factors computed by SSYTRD in reducing A to tridiagonal form.
  • Z — REAL array of dimension( LDU, max(NN) ). The orthogonal matrix of eigenvectors computed by SSTEQR, SPTEQR, and SSTEIN.
  • WORK — REAL array of dimension( LWORK ). Workspace. That's it. That's the description 🫶.
  • LWORK — INTEGER. The number of entries in WORK. This must be at least
1 + 4 * Nmax + 2 * Nmax * lg Nmax + 3 * Nmax**2

where Nmax = max( NN(j), 2 ) and lg = log base 2. Do not skimp on this one, bestie — it's not a vibe check, it's a hard requirement 😭.

  • IWORK — INTEGER array, Workspace. The integer-flavoured workspace, obviously.
  • LIWORK — INTEGER. The number of entries in IWORK. This must be at least
6 + 6*Nmax + 5 * Nmax * lg Nmax

where Nmax = max( NN(j), 2 ) and lg = log base 2. Same deal, same love 💖.

  • RESULT — REAL array, dimension (26). The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. Twenty-six slots of pure test output — a whole mood board 🌈.
  • INFO —
  • INFO is INTEGER — the run’s main-character report card 💅

If it’s 0, then everything ran OK. If it’s negative, it’s clocking what went wrong:

  • -1: NSIZES < 0
  • -2: Some NN(j) < 0
  • -3: NTYPES < 0
  • -5: THRESH < 0
  • -9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ).
  • -23: LDU < 1 or LDU < NMAX.
  • -29: LWORK too small.

And if SLATMR, SLATMS, SSYTRD, SORGTR, SSTEQR, SSTERF, or SORMC2 returns an error code, the absolute value of it is returned. Not me taking the absolute value so the sign drama doesn’t leak 😭

🧚 Some Local Variables and Parameters 🧚

  • ZERO, ONE — Real 0 and 1. iconic basics.
  • 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.
  • NBLOCK — blocksize as returned by ENVIR.
  • NMAX — largest value in NN.
  • NMATS — the number of matrices generated so far.
  • NERRS — the number of tests which have exceeded THRESH so far.
  • 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 — finest relative precision and its inverse.
  • RTOVFL, RTUNFL — square roots of the previous 2 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) ) )

🎀 Author 🎀

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

Definition at line 587 of file schkst.f.

🎀 Author 🎀

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