TESTING/EIG/zdrvvx.f(3)

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🌸 TESTING/EIG/zdrvvx.f(3) — Library Functions Manual

okay bestie, welcome to the manual page for TESTING/EIG/zdrvvx.f from the Library Functions Manual 💖 This is the ZDRVVX tester, and it is genuinely about to check the nonsymmetric eigenvalue problem expert driver ZGEEVX. Slay.

💅 NAME

TESTING/EIG/zdrvvx.f — the ZDRVVX routine, aka the tester that puts ZGEEVX through its paces and makes sure the nonsymmetric eigenvalue tea is not spilling everywhere 🫶

✨ SYNOPSIS

🎀 Functions/Subroutines

  • subroutine zdrvvx (...) — ZDRVVX
subroutine zdrvvx (nsizes, nn, ntypes, dotype, iseed, thresh, niunit,
    nounit, a, lda, h, w, w1, vl, ldvl, vr, ldvr, lre, ldlre, rcondv,
    rcndv1, rcdvin, rconde, rcnde1, rcdein, scale, scale1, result, work,
    nwork, rwork, info)
    ZDRVVX

💖 Function/Subroutine Documentation

🦋 ZDRVVX

subroutine zdrvvx (integer nsizes, integer, dimension( * ) nn, integer
    ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed,
    double precision thresh, integer niunit, integer nounit, complex*16,
    dimension( lda, * ) a, integer lda, complex*16, dimension( lda, * ) h,
    complex*16, dimension( * ) w, complex*16, dimension( * ) w1, complex*16,
    dimension( ldvl, * ) vl, integer ldvl, complex*16, dimension( ldvr, * )
    vr, integer ldvr, complex*16, dimension( ldlre, * ) lre, integer ldlre,
    double precision, dimension( * ) rcondv, double precision, dimension( *
    ) rcndv1, double precision, dimension( * ) rcdvin, double precision,
    dimension( * ) rconde, double precision, dimension( * ) rcnde1, double
    precision, dimension( * ) rcdein, double precision, dimension( * )
    scale, double precision, dimension( * ) scale1, double precision,
    dimension( 11 ) result, complex*16, dimension( * ) work, integer nwork,
    double precision, dimension( * ) rwork, integer info)
    ZDRVVX

🌟 Purpose

okay so ZDRVVX checks the nonsymmetric eigenvalue problem expert driver ZGEEVX 👑 no cap.

ZDRVVX uses both test matrices generated randomly depending on data supplied in the calling sequence, and data read from an input file, including precomputed condition numbers to which it compares the ones it computes. It’s giving “I brought my own answer key” energy 💅

When ZDRVVX is called, a number of matrix sizes and a number of matrix types are specified in the calling sequence. For each size and each type of matrix, one matrix will be generated and used to test the nonsymmetric eigenroutines. For each matrix, 9 tests will be performed:

  1. (1) | A * VR - VR * W | / ( n |A| ulp )

Here VR is the matrix of unit right eigenvectors. W is a diagonal matrix with diagonal entries W(j).

  1. (2) | A**H * VL - VL * W**H | / ( n |A| ulp )

Here VL is the matrix of unit left eigenvectors, A**H is the conjugate transpose of A, and W is as above.

  1. (3) | |VR(i)| - 1 | / ulp and largest component real

VR(i) denotes the i-th column of VR.

  1. (4) | |VL(i)| - 1 | / ulp and largest component real

VL(i) denotes the i-th column of VL.

  1. (5) W(full) = W(partial)

W(full) denotes the eigenvalues computed when VR, VL, RCONDV and RCONDE are also computed, and W(partial) denotes the eigenvalues computed when only some of VR, VL, RCONDV, and RCONDE are computed.

  1. (6) VR(full) = VR(partial)

VR(full) denotes the right eigenvectors computed when VL, RCONDV and RCONDE are computed, and VR(partial) denotes the result when only some of VL and RCONDV are computed.

  1. (7) VL(full) = VL(partial)

VL(full) denotes the left eigenvectors computed when VR, RCONDV and RCONDE are computed, and VL(partial) denotes the result when only some of VR and RCONDV are computed.

  1. (8) 0 if SCALE, ILO, IHI, ABNRM (full) = SCALE, ILO, IHI, ABNRM (partial); 1/ulp otherwise

SCALE, ILO, IHI and ABNRM describe how the matrix is balanced. (full) is when VR, VL, RCONDE and RCONDV are also computed, and (partial) is when some are not computed.

  1. (9) RCONDV(full) = RCONDV(partial)

RCONDV(full) denotes the reciprocal condition numbers of the right eigenvectors computed when VR, VL and RCONDE are also computed. RCONDV(partial) denotes the reciprocal condition numbers when only some of VR, VL and RCONDE are computed.

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 complex angles. (ULP = (first number larger than 1) - 1)
  5. A diagonal matrix with geometrically spaced entries 1, ..., ULP and random complex angles.
  6. A diagonal matrix with entries 1, ULP, ..., ULP and random complex angles.
  7. Same as (4), but multiplied by a constant near the overflow threshold.
  8. Same as (4), but multiplied by a constant near the underflow threshold.
  9. A matrix of the form U' T U, where U is unitary and T has evenly spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle.
  10. A matrix of the form U' T U, where U is unitary and T has geometrically spaced entries 1, ..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle.
  11. A matrix of the form U' T U, where U is unitary and T has entries 1, ULP,..., ULP with random complex angles on the diagonal and random O(1) entries in the upper triangle.
  12. A matrix of the form U' T U, where U is unitary and T has complex eigenvalues randomly chosen from ULP < |z| < 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 complex angles 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 complex angles 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 complex angles 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 complex eigenvalues randomly chosen from ULP < |z| < 1 and random O(1) entries in the upper triangle.
  17. Same as (16), but multiplied by a constant near the overflow threshold.
  18. Same as (16), but multiplied by a constant near the underflow threshold.
  19. Nonsymmetric matrix with random entries chosen from |z| < 1. If N is at least 4, all entries in first two rows and last row, and first column and last two columns are zero.
  20. Same as (19), but multiplied by a constant near the overflow threshold.
  21. Same as (19), but multiplied by a constant near the underflow threshold.

In addition, an input file will be read from logical unit number NIUNIT. The file contains matrices along with precomputed eigenvalues and reciprocal condition numbers for the eigenvalues and right eigenvectors. For these matrices, in addition to tests (1) to (9) we will compute the following two tests:

  • (10) |RCONDV - RCDVIN| / cond(RCONDV)

RCONDV is the reciprocal right eigenvector condition number computed by ZGEEVX and RCDVIN (the precomputed true value) is supplied as input. cond(RCONDV) is the condition number of RCONDV, and takes errors in computing RCONDV into account, so that the resulting quantity should be O(ULP). cond(RCONDV) is essentially given by norm(A)/RCONDE.

  • (11) |RCONDE - RCDEIN| / cond(RCONDE)

RCONDE is the reciprocal eigenvalue condition number computed by ZGEEVX and RCDEIN (the precomputed true value) is supplied as input. cond(RCONDE) is the condition number of RCONDE, and takes errors in computing RCONDE into account, so that the resulting quantity should be O(ULP). cond(RCONDE) is essentially given by norm(A)/RCONDV.

🌈 Parameters

NSIZES

NSIZES is INTEGER — the number of sizes of matrices to use. It must be at least zero. If it is zero, no randomly generated matrices are tested, but any test matrices read from NIUNIT will be tested. ✨

NN

NN is 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. 🎀

NTYPES

NTYPES is INTEGER — the number of elements in DOTYPE. It must be at least zero. If it is zero, no randomly generated test matrices are tested, but any test matrices read from NIUNIT will be tested. 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.. 💅

DOTYPE

DOTYPE

DOTYPE is a 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 biggest number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP just do not get generated. And if NTYPES is bigger than MAXTYP, then DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) are ignored. That's it. 💅

ISEED

ISEED is an INTEGER array, dimension (4).

On entry, ISEED sets the seed of the random number generator. The array elements should be between 0 and 4095 — if they're not, they get reduced mod 4096. Also ISEED(4) must be odd. The RNG 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 use them in the next call to ZDRVVX to continue the same random-number sequence. Slay, continuity. ✨

THRESH

THRESH is DOUBLE PRECISION.

A test counts as a fail if the ratio — 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 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. No cap. 🫶

NIUNIT

NIUNIT is INTEGER.

It's the FORTRAN unit number for reading in the data file of problems to solve. That's the tea. 🍓

NOUNIT

NOUNIT is INTEGER.

It's the FORTRAN unit number for printing out error messages — like if a routine returns INFO not equal to 0. 😭

A

A is a COMPLEX*16 array, dimension (LDA, max(NN,12)).

It holds the matrix whose eigenvalues are gonna be computed. On exit, A contains the last matrix actually used. Main character matrix energy. 👑

LDA

LDA is INTEGER.

The leading dimension of A and H. LDA must be at least 1 and at least max( NN, 12 ). (12 is the dimension of the largest matrix on the precomputed input file.) Lowkey important. ⭐

H

H is a COMPLEX*16 array, dimension (LDA, max(NN,12)).

Another copy of the test matrix A, modified by ZGEEVX. 🦋

W

W is a COMPLEX*16 array, dimension (max(NN,12)).

Contains the eigenvalues of A. 💜

W1

W1 is a COMPLEX*16 array, dimension (max(NN,12)).

Like W, this array contains the eigenvalues of A — but those computed when ZGEEVX only computes a partial eigendecomposition, i.e. not the eigenvalues and left and right eigenvectors. Partial slay. 🌸

VL

VL is a COMPLEX*16 array, dimension (LDVL, max(NN,12)).

VL holds the computed left eigenvectors. 💫

LDVL

LDVL is INTEGER.

Leading dimension of VL. Must be at least max(1,max(NN,12)). 🎀

VR

VR is a COMPLEX*16 array, dimension (LDVR, max(NN,12)).

VR holds the computed right eigenvectors. 🔥

LDVR

LDVR is INTEGER.

Leading dimension of VR. Must be at least max(1,max(NN,12)). 🧚

LRE

LRE is a COMPLEX*16 array, dimension (LDLRE, max(NN,12)).

LRE holds the computed right or left eigenvectors. Either way, it's booked. 💅

LDLRE

LDLRE is INTEGER.

Leading dimension of LRE. Must be at least max(1,max(NN,12)). 🌈

RCONDV

RCONDV is a DOUBLE PRECISION array, dimension (N).

RCONDV holds the computed reciprocal condition numbers for eigenvectors. 🎯

RCNDV1

RCNDV1 is a DOUBLE PRECISION array, dimension (N).

RCNDV1 holds more computed reciprocal condition numbers for eigenvectors. More numbers, more vibes. ✨

RCDVIN

RCDVIN is a DOUBLE PRECISION array, dimension (N).

When COMP = .TRUE., RCDVIN holds the precomputed reciprocal condition numbers for eigenvectors to be compared with RCONDV. 👀

RCONDE

RCONDE is a DOUBLE PRECISION array, dimension (N).

RCONDE holds the computed reciprocal condition numbers for eigenvalues. 🧮

RCNDE1

RCNDE1 is a DOUBLE PRECISION array, dimension (N).

RCNDE1 holds more computed reciprocal condition numbers for eigenvalues. Again: more. 😌

RCDEIN

RCDEIN is a DOUBLE PRECISION array, dimension (N).

When COMP = .TRUE., RCDEIN holds the precomputed reciprocal condition numbers for eigenvalues to be compared with RCONDE. 💖

SCALE

SCALE is a DOUBLE PRECISION array, dimension (N).

Holds information describing balancing of the matrix. ⚖️

SCALE1

SCALE1 is a DOUBLE PRECISION array, dimension (N).

Holds information describing balancing of the matrix. Yes, again. It's giving backup balance sheet. 🎀

WORK

WORK is a COMPLEX*16 array, dimension (NWORK).

It works. (That's the whole description. Respect.) 💅

RESULT

RESULT is a DOUBLE PRECISION array, dimension (11).

The values computed by the seven tests described above. The values are currently limited to 1/ulp to avoid overflow. Safety first, bestie. 🛟

NWORK

NWORK is INTEGER.

The number of entries in WORK. This must be at least:

max(6*12+2*12**2,6*NN(j)+2*NN(j)**2) =
max(    360     ,6*NN(j)+2*NN(j)**2)    for all j.

No shrinking allowed. 📏

RWORK

RWORK is a DOUBLE PRECISION array, dimension (2*max(NN,12)).

Real work, real values. 💜

INFO

INFO is INTEGER.

  • If 0, then successful exit. 🎉
  • If <0, then input parameter -INFO is incorrect. 🙈
  • If >0, then ZLATMR, CLATMS, CLATME or ZGET23 returned an error code, and INFO is its absolute value. 💀

🧚 Some Local Variables and Parameters

The locals, decoded

  • ZERO, ONE — Real 0 and 1.
  • MAXTYP — The number of types defined.
  • NMAX — Largest value in NN or 12.
  • NERRS — The number of tests which have exceeded THRESH.
  • 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.
  • RTULP, RTULPI — Square roots of the previous 4 values. Math girlies rise. 🧮✨

The four arrays that 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) — Selectw whether CONDS is to be 1 or 1/sqrt(ulp). (0 means irrelevant.) (Yes, "Selectw" is in the source. We don't fix, we appreciate. 💅)

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 491 of file zdrvvx.f.

🦋 Author

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