DDRVST2STG(3)

📦
version 3.12.1 · 2 versions
, in arch rolling

💖 ddrvst2stg — TESTING/EIG/ddrvst2stg.f 💖

okay bestie, welcome to the (3) manual page for TESTING/EIG/ddrvst2stg.f, straight out of the Library Functions Manual. This is the symmetric-eigenvalue-driver test routine, and she is lowkey the main character of the testing folder. 🦋

✨ NAME ✨

TESTING/EIG/ddrvst2stg.f

🌸 SYNOPSIS 🌸

🎀 Functions/Subroutines 🎀

subroutine ddrvst2stg (nsizes, nn, ntypes, dotype, iseed, thresh,
    nounit, a, lda, d1, d2, d3, d4, eveigs, wa1, wa2, wa3, u, ldu, v,
    tau, z, work, lwork, iwork, liwork, result, info)

DDRVST2STG

🦋 Function/Subroutine Documentation 🦋

✨ subroutine ddrvst2stg full signature ✨

subroutine ddrvst2stg (integer nsizes, integer, dimension( * ) nn, integer
    ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed,
    double precision thresh, integer nounit, double precision, dimension(
    lda, * ) a, integer lda, double precision, dimension( * ) d1, double
    precision, dimension( * ) d2, double precision, dimension( * ) d3,
    double precision, dimension( * ) d4, double precision, dimension( * )
    eveigs, double precision, dimension( * ) wa1, double precision,
    dimension( * ) wa2, double precision, dimension( * ) wa3, double
    precision, dimension( ldu, * ) u, integer ldu, double precision,
    dimension( ldu, * ) v, double precision, dimension( * ) tau, double
    precision, dimension( ldu, * ) z, double precision, dimension( * ) work,
    integer lwork, integer, dimension( * ) iwork, integer liwork, double
    precision, dimension( * ) result, integer info)
DDRVST2STG

💅 Purpose: 💅

DDRVST2STG checks the symmetric eigenvalue problem drivers. Like, it is the vibe check for all of these:

  • DSTEV — computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix.
  • DSTEVX — computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix.
  • DSTEVR — computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix using the Relatively Robust Representation where it can.
  • DSYEV — computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix.
  • DSYEVX — computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix.
  • DSYEVR — computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix using the Relatively Robust Representation where it can.
  • DSPEV — computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix in packed storage.
  • DSPEVX — computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix in packed storage.
  • DSBEV — computes all eigenvalues and, optionally, eigenvectors of a real symmetric band matrix.
  • DSBEVX — computes selected eigenvalues and, optionally, eigenvectors of a real symmetric band matrix.
  • DSYEVD — computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix using a divide and conquer algorithm.
  • DSPEVD — computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix in packed storage, using a divide and conquer algorithm.
  • DSBEVD — computes all eigenvalues and, optionally, eigenvectors of a real symmetric band matrix, using a divide and conquer algorithm.
bestie, the source leaves a couple of blank spots in this text, so I’m marking them as [blank] and not inventing facts. 🫶

When DDRVST2STG is called, a number of matrix [blank] (()) and a number of matrix [blank] are specified. For each size (()) and each type of matrix, one matrix will be generated and used to test the appropriate drivers. For each matrix and each driver routine called, the following tests will be performed:

  1. | A - Z D Z' | / ( |A| n ulp )
  2. | I - Z Z' | / ( n ulp )
  3. | D1 - D2 | / ( |D1| ulp )

where Z is the matrix of eigenvectors returned when the eigenvector option is given and D1 and D2 are the eigenvalues returned with and without the eigenvector option.

The [blank] are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The [blank] are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type [blank] 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 eigenvalues 1, ..., ULP [blank] and random signs. (ULP = (first number larger than 1) - 1)
  • (4) A diagonal matrix with geometrically spaced eigenvalues 1, ..., ULP [blank] and random signs.
  • (5) A diagonal matrix with [blank] eigenvalues 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 [blank] 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) A band matrix with half bandwidth randomly chosen between 0 and N-1, with evenly spaced eigenvalues 1, ..., ULP with random signs.
  • (17) Same as (16), but multiplied by SQRT( overflow threshold )
  • (18) Same as (16), but multiplied by SQRT( underflow threshold )

okay so we're still deep in the argument list and honestly? it's giving "professor who loves details" and we respect it. let's go through every single one, no skipping, periodt. 💅

  • NSIZES (INTEGER) — the number of sizes of matrices to use. if it's zero, DDRVST2STG literally does nothing (like, no thoughts, head empty). it must be at least zero. Not modified.
  • NN (INTEGER array, dimension (NSIZES)) — an array holding the sizes to be used for the matrices. zero values will be skipped (we don't stan zero-sized matrices here). the values must be at least zero. Not modified.
  • NTYPES (INTEGER) — the number of elements in DOTYPE. if it's zero, DDRVST2STG 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 (no even numbers allowed, bestie). 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 DDRVST2STG to continue the same random number sequence. Modified.
  • THRESH (DOUBLE PRECISION) — a test will count as if the , computed as described above, exceeds THRESH. (yes, that sentence is exactly what the original says, we don't make the rules 😭). 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 (DOUBLE PRECISION 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. it must be at least 1 and at least max( NN ). Not modified.
  • D1 (DOUBLE PRECISION array, dimension (max(NN))) — the eigenvalues of A, as computed by DSTEQR simultaneously with Z. on exit, the eigenvalues in D1 correspond with the matrix in A. Modified.
  • D2 (DOUBLE PRECISION array, dimension (max(NN))) — the eigenvalues of A, as computed by DSTEQR if Z is not computed. on exit, the eigenvalues in D2 correspond with the matrix in A. Modified.
  • D3 (DOUBLE PRECISION array, dimension (max(NN))) — the eigenvalues of A, as computed by DSTERF. on exit, the eigenvalues in D3 correspond with the matrix in A. Modified.
  • D4 (DOUBLE PRECISION array, dimension) — (no description, it's literally just vibes 💀).
  • EVEIGS (DOUBLE PRECISION array, dimension (max(NN))) — the eigenvalues as computed by DSTEV('N', ... ) (I reserve the right to change this to the output of whichever algorithm computes the most accurate eigenvalues). (yes, the original author left that note in, iconic).
  • WA1 (DOUBLE PRECISION array, dimension) — (no description, mysterious).
  • WA2 (DOUBLE PRECISION array, dimension) — (no description, also mysterious).
  • WA3 (DOUBLE PRECISION array, dimension) — (no description, the mystery continues).
  • U (DOUBLE PRECISION array, dimension (LDU, max(NN))) — the orthogonal matrix computed by DSYTRD + DORGTR. Modified.
  • LDU (INTEGER) — the leading dimension of U, Z, and V. it must be at least 1 and at least max( NN ). Not modified.
  • V (DOUBLE PRECISION array, dimension (LDU, max(NN))) — the Housholder vectors computed by DSYTRD in reducing A to tridiagonal form. (yes, "Housholder" with one 'e' — we're not fixing it, we're keeping the original spelling). Modified.
  • TAU (DOUBLE PRECISION array, dimension (max(NN))) — the Householder factors computed by DSYTRD in reducing A to tridiagonal form. Modified.
  • Z (DOUBLE PRECISION array, dimension (LDU, max(NN))) — the orthogonal matrix of eigenvectors computed by DSTEQR, DPTEQR, and DSTEIN. Modified.
  • WORK (DOUBLE PRECISION array, dimension (LWORK)) — workspace. Modified.
  • LWORK (INTEGER) — the number of entries in WORK. this must be at least
1 + 4 * Nmax + 2 * Nmax * lg Nmax + 4 * Nmax**2

where Nmax = max( NN(j), 2 ) and lg = log base 2. Not modified.

  • IWORK (INTEGER array, dimension (6 + 6Nmax + 5 * Nmax * lg Nmax )*) — where Nmax = max( NN(j), 2 ) and lg = log base 2. workspace. Modified.
  • RESULT (DOUBLE PRECISION array, dimension (105)) — the values computed by the tests described above. the values are currently limited to 1/ulp, to avoid overflow. Modified.
  • INFO (INTEGER) — if 0, then everything ran OK. otherwise:
    • -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) ).
    • -16: LDU < 1 or LDU < NMAX.
    • -21: LWORK too small. If DLATMR, DLATMS, DSYTRD, DORGTR, DSTEQR, DSTERF, or DORMTR returns an error code, the absolute value of it is returned. Modified.

now, the local variables and parameters? literally the backstage crew. here's who's who:

  • 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.
  • 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, 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) )

and finally, the tests performed are: (yes, this is the whole mood board, Routine tested is on the right)

1= | A - U S U' | / ( |A| n ulp )         DSTEV('V', ... )
2= | I - U U' | / ( n ulp )               DSTEV('V', ... )
3= |D(with Z) - D(w/o Z)| / (|D| ulp)     DSTEV('N', ... )
4= | A - U S U' | / ( |A| n ulp )         DSTEVX('V','A', ... )
5= | I - U U' | / ( n ulp )               DSTEVX('V','A', ... )
6= |D(with Z) - EVEIGS| / (|D| ulp)       DSTEVX('N','A', ... )
7= | A - U S U' | / ( |A| n ulp )         DSTEVR('V','A', ... )
8= | I - U U' | / ( n ulp )               DSTEVR('V','A', ... )
9= |D(with Z) - EVEIGS| / (|D| ulp)       DSTEVR('N','A', ... )
10= | A - U S U' | / ( |A| n ulp )        DSTEVX('V','I', ... )
11= | I - U U' | / ( n ulp )              DSTEVX('V','I', ... )
12= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSTEVX('N','I', ... )
13= | A - U S U' | / ( |A| n ulp )        DSTEVX('V','V', ... )
14= | I - U U' | / ( n ulp )              DSTEVX('V','V', ... )
15= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSTEVX('N','V', ... )
16= | A - U S U' | / ( |A| n ulp )        DSTEVD('V', ... )
17= | I - U U' | / ( n ulp )              DSTEVD('V', ... )
18= |D(with Z) - EVEIGS| / (|D| ulp)      DSTEVD('N', ... )
19= | A - U S U' | / ( |A| n ulp )        DSTEVR('V','I', ... )
20= | I - U U' | / ( n ulp )              DSTEVR('V','I', ... )
21= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSTEVR('N','I', ... )
22= | A - U S U' | / ( |A| n ulp )        DSTEVR('V','V', ... )
23= | I - U U' | / ( n ulp )              DSTEVR('V','V', ... )
24= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSTEVR('N','V', ... )

25= | A - U S U' | / ( |A| n ulp )        DSYEV('L','V', ... )
26= | I - U U' | / ( n ulp )              DSYEV('L','V', ... )
27= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSYEV_2STAGE('L','N', ... )

okay so we’re picking up mid-list at test 28, bestie — no cap, the rest of the checklist is right here 💅

28= | A - U S U' | / ( |A| n ulp )        DSYEVX('L','V','A', ... )
29= | I - U U' | / ( n ulp )              DSYEVX('L','V','A', ... )
30= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSYEVX_2STAGE('L','N','A', ... )
31= | A - U S U' | / ( |A| n ulp )        DSYEVX('L','V','I', ... )
32= | I - U U' | / ( n ulp )              DSYEVX('L','V','I', ... )
33= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSYEVX_2STAGE('L','N','I', ... )
34= | A - U S U' | / ( |A| n ulp )        DSYEVX('L','V','V', ... )
35= | I - U U' | / ( n ulp )              DSYEVX('L','V','V', ... )
36= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSYEVX_2STAGE('L','N','V', ... )
37= | A - U S U' | / ( |A| n ulp )        DSPEV('L','V', ... )
38= | I - U U' | / ( n ulp )              DSPEV('L','V', ... )
39= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSPEV('L','N', ... )
40= | A - U S U' | / ( |A| n ulp )        DSPEVX('L','V','A', ... )
41= | I - U U' | / ( n ulp )              DSPEVX('L','V','A', ... )
42= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSPEVX('L','N','A', ... )
43= | A - U S U' | / ( |A| n ulp )        DSPEVX('L','V','I', ... )
44= | I - U U' | / ( n ulp )              DSPEVX('L','V','I', ... )
45= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSPEVX('L','N','I', ... )
46= | A - U S U' | / ( |A| n ulp )        DSPEVX('L','V','V', ... )
47= | I - U U' | / ( n ulp )              DSPEVX('L','V','V', ... )
48= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSPEVX('L','N','V', ... )
49= | A - U S U' | / ( |A| n ulp )        DSBEV('L','V', ... )
50= | I - U U' | / ( n ulp )              DSBEV('L','V', ... )
51= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSBEV_2STAGE('L','N', ... )
52= | A - U S U' | / ( |A| n ulp )        DSBEVX('L','V','A', ... )
53= | I - U U' | / ( n ulp )              DSBEVX('L','V','A', ... )
54= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSBEVX_2STAGE('L','N','A', ... )
55= | A - U S U' | / ( |A| n ulp )        DSBEVX('L','V','I', ... )
56= | I - U U' | / ( n ulp )              DSBEVX('L','V','I', ... )
57= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSBEVX_2STAGE('L','N','I', ... )
58= | A - U S U' | / ( |A| n ulp )        DSBEVX('L','V','V', ... )
59= | I - U U' | / ( n ulp )              DSBEVX('L','V','V', ... )
60= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSBEVX_2STAGE('L','N','V', ... )
61= | A - U S U' | / ( |A| n ulp )        DSYEVD('L','V', ... )
62= | I - U U' | / ( n ulp )              DSYEVD('L','V', ... )
63= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSYEVD_2STAGE('L','N', ... )
64= | A - U S U' | / ( |A| n ulp )        DSPEVD('L','V', ... )
65= | I - U U' | / ( n ulp )              DSPEVD('L','V', ... )
66= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSPEVD('L','N', ... )
67= | A - U S U' | / ( |A| n ulp )        DSBEVD('L','V', ... )
68= | I - U U' | / ( n ulp )              DSBEVD('L','V', ... )
69= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSBEVD_2STAGE('L','N', ... )
70= | A - U S U' | / ( |A| n ulp )        DSYEVR('L','V','A', ... )
71= | I - U U' | / ( n ulp )              DSYEVR('L','V','A', ... )
72= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSYEVR_2STAGE('L','N','A', ... )
73= | A - U S U' | / ( |A| n ulp )        DSYEVR('L','V','I', ... )
74= | I - U U' | / ( n ulp )              DSYEVR('L','V','I', ... )
75= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSYEVR_2STAGE('L','N','I', ... )
76= | A - U S U' | / ( |A| n ulp )        DSYEVR('L','V','V', ... )
77= | I - U U' | / ( n ulp )              DSYEVR('L','V','V', ... )
78= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSYEVR_2STAGE('L','N','V', ... )

Quick note from the source: Tests 25 through 78 are repeated (as tests 79 through 132) with UPLO='U'. Iconic recycling, honestly ♻️✨

Then it literally says To be added in 1999 — yes, a whole time capsule — and lists:

79= | A - U S U' | / ( |A| n ulp )        DSPEVR('L','V','A', ... )
80= | I - U U' | / ( n ulp )              DSPEVR('L','V','A', ... )
81= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSPEVR('L','N','A', ... )
82= | A - U S U' | / ( |A| n ulp )        DSPEVR('L','V','I', ... )
83= | I - U U' | / ( n ulp )              DSPEVR('L','V','I', ... )
84= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSPEVR('L','N','I', ... )
85= | A - U S U' | / ( |A| n ulp )        DSPEVR('L','V','V', ... )
86= | I - U U' | / ( n ulp )              DSPEVR('L','V','V', ... )
87= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSPEVR('L','N','V', ... )
88= | A - U S U' | / ( |A| n ulp )        DSBEVR('L','V','A', ... )
89= | I - U U' | / ( n ulp )              DSBEVR('L','V','A', ... )
90= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSBEVR('L','N','A', ... )
91= | A - U S U' | / ( |A| n ulp )        DSBEVR('L','V','I', ... )
92= | I - U U' | / ( n ulp )              DSBEVR('L','V','I', ... )
93= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSBEVR('L','N','I', ... )
94= | A - U S U' | / ( |A| n ulp )        DSBEVR('L','V','V', ... )
95= | I - U U' | / ( n ulp )              DSBEVR('L','V','V', ... )
96= |D(with Z) - D(w/o Z)| / (|D| ulp)    DSBEVR('L','N','V', ... )

Author 👑

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

Definition at line 449 of file ddrvst2stg.f.

Author ✨

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That’s the whole page, bestie — shoutout to LAPACK Version 3.12.0 and TESTING/EIG/ddrvst2stg.f(3), with the footer date doing its tiny little cameo at the very bottom. 💖✨🫶

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