SCHKST(3)

📦 version 3.12.0, in opensuse-leap 16.0

🌸 NAME 🌸

okay bestie, so this page is from the Library Functions Manual and it's all about schkst.f(3) — like, the whole entire path is:

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

that's literally it, that's the name, it's giving "I live in a build directory" energy 💅

✨ 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) — yes queen, this is the one, the whole cast list 🫶
    • SCHKST

🎀 Function/Subroutine Documentation 🎀

💖 subroutine SCHKST 💖

okay so here's the full signature, and no I did not skip a single argument, we are NOT doing that today:

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 listen up, because this is the whole lore 🍓

SCHKST checks the symmetric eigenvalue problem routines. iconique behaviour, truly 👑

SSYTRD factors A as U S U' , where ' means transpose, S is symmetric tridiagonal, and U is orthogonal. and yes, SSYTRD can use either just the lower or just the upper triangle of A — SCHKST checks both cases because we're thorough like that 💅

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. (that's literally what it says bestie, the word before "format" is just… vibes 🌸)

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. (no eigenvectors, no problem 😌)

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. highkey a flex 💪

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). she does it all, main character energy fr fr ✨

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. RRR?? she's so real for that 🦋

When SCHKST 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 symmetric eigenroutines. For each matrix, a number of tests will be performed:

  • (1) | A - V S V' | / ( |A| n ulp ) — SSYTRD( UPLO='U', ... )
  • (2) | I - UV' | / ( n ulp ) — SORGTR( UPLO='U', ... )
  • (3) | A - V S V' | / ( |A| n ulp ) — SSYTRD( UPLO='L', ... )
  • (4) | I - UV' | / ( n ulp ) — SORGTR( UPLO='L', ... )
  • (5-8) Same as 1-4, but for SSPTRD and SOPGTR. (a little sequel, we love a franchise 🎬)
  • (9) | S - Z D Z' | / ( |S| n ulp ) — SSTEQR('V',...)
  • (10) | I - ZZ' | / ( n ulp ) — SSTEQR('V',...)
  • (11) | D1 - D2 | / ( |D1| ulp ) — SSTEQR('N',...)
  • (12) | D1 - D3 | / ( |D1| ulp ) — 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 ) — SPTEQR('V',...)
  • (15) | I - Z4 Z4' | / ( n ulp ) — SPTEQR('V',...)
  • (16) | D4 - D5 | / ( 100 |D4| ulp ) — SPTEQR('N',...)

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

(17) the omega moment:

(17)    max | D4(i) - WR(i) | / ( |D4(i)| omega ) ,
         i
        omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
                                             SSTEBZ( 'A', 'E', ...)
  • (18) | WA1 - D3 | / ( |D3| ulp ) — SSTEBZ( 'A', 'E', ...)

(19) the min/max mashup:

(19)    ( max { min | WA2(i)-WA3(j) | } +
           i     j
          max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
           i     j
                                             SSTEBZ( 'I', 'E', ...)
  • (20) | S - Y WA1 Y' | / ( |S| n ulp ) — SSTEBZ, SSTEIN
  • (21) | I - Y Y' | / ( n ulp ) — SSTEBZ, SSTEIN
  • (22) | S - Z D Z' | / ( |S| n ulp ) — SSTEDC('I')
  • (23) | I - ZZ' | / ( n ulp ) — SSTEDC('I')
  • (24) | S - Z D Z' | / ( |S| n ulp ) — SSTEDC('V')
  • (25) | I - ZZ' | / ( n ulp ) — SSTEDC('V')
  • (26) | D1 - D2 | / ( |D1| ulp ) — SSTEDC('V') and SSTEDC('N')
🫶 heads up bestie: Test 27 is disabled at the moment because SSTEMR does not guarantee high relatvie accuracy. (yes, "relatvie" — I'm not fixing it, that's the original spelling, we respect the source material 💀)

(27) another omega situation:

(27)    max | D6(i) - WR(i) | / ( |D6(i)| omega ) ,
         i
        omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
                                             SSTEMR('V', 'A')

(28) and its twin:

(28)    max | D6(i) - WR(i) | / ( |D6(i)| omega ) ,
         i
        omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
                                             SSTEMR('V', 'I')
💀 Tests 29 through 34 are disable at present because SSTEMR does not handle partial spectrum requests. (again — "disable", keeping it exactly as given, no edits from me 😭)
  • (29) | S - Z D Z' | / ( |S| n ulp ) — SSTEMR('V', 'I')
  • (30) | I - ZZ' | / ( n ulp ) — SSTEMR('V', 'I')

(31) the comparison era:

(31)    ( max { min | WA2(i)-WA3(j) | } +
           i     j
          max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
           i     j
        SSTEMR('N', 'I') vs. SSTEMR('V', 'I')
  • (32) | S - Z D Z' | / ( |S| n ulp ) — SSTEMR('V', 'V')
  • (33) | I - ZZ' | / ( n ulp ) — SSTEMR('V', 'V')

(34) round two of the mashup:

(34)    ( max { min | WA2(i)-WA3(j) | } +
           i     j
          max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
           i     j
        SSTEMR('N', 'V') vs. SSTEMR('V', 'V')
  • (35) | S - Z D Z' | / ( |S| n ulp ) — SSTEMR('V', 'A')
  • (36) | I - ZZ' | / ( n ulp ) — SSTEMR('V', 'A')

(37) and the grand finale of comparisons:

(37)    ( max { min | WA2(i)-WA3(j) | } +
           i     j
          max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
           i     j
        SSTEMR('N', 'A') vs. SSTEMR('V', 'A')

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 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
  • NSIZES — INTEGER, bestie. This is the number of sizes of matrices you're gonna use. If it's zero, SCHKST does nothing — like, absolutely nothing, no cap. It must be at least zero. 💅
  • NN — INTEGER array, dimension (NSIZES). This array holds the sizes used for the matrices. Zero values get skipped. The values must be at least zero. ✨
  • NTYPES — INTEGER. The number of elements in DOTYPE. If it's zero, SCHKST does nothing. It must be at least zero. If it's 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.. 🫶
  • 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. 👑
  • 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 SCHKST to continue the same random number sequence. 🌈
  • THRESH — REAL. A test will count as ___ if the ___ (the original literally has blanks here 😭), 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. 💖
  • NOUNIT — INTEGER. The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.) 🎀
  • A — REAL array of dimension ( LDA , max(NN) ). Used to hold the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used. ✨
  • LDA — INTEGER. The leading dimension of A. It must be at least 1 and at least max( NN ). 💅
  • AP — REAL array of dimension( max(NN)*max(NN+1)/2 ). The matrix A stored in packed format. 🦋
  • 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. 💜
  • 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 here — not a typo from me 😭.) 💀
  • 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. 🍓
  • 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. 💖
  • 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 ). 💅
  • V — REAL array of dimension( LDU, max(NN) ). The Housholder 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.) ✨
  • VP — REAL array of dimension( max(NN)*max(NN+1)/2 ). The matrix V stored in packed format. 🎀
  • 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 ). 🫶
  • 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. 🔥
  • IWORK — INTEGER array, Workspace. ⭐
  • 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. 🌈
  • RESULT — REAL array, dimension (26). The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. 👑
  • INFO — the page just lists INFO and then dips 😭. There's no description text after it in this excerpt. 🙈

INFO is an INTEGER, bestie, and it’s literally the vibe check for the whole run 👀. Here’s how to read it:

  • 0 — everything ran OK, slay ✅
  • -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.

If SLATMR, SLATMS, SSYTRD, SORGTR, SSTEQR, SSTERF, or SORMC2 returns an error code, the absolute value of it is returned. iykyk 🫶

Some Local Variables and Parameters 🔍

Okay so here’s the supporting cast, no cap:

  • ZERO, ONE — Real 0 and 1.
  • 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 👑

Shoutout to the legends behind this:

  • 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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This page is LAPACK Version 3.12.0, sourced from /home/abui...IG/schkst.f(3) — catch you later, bestie 💖✨🦋

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