ZDRVVX(3)

📦 version 3.12.0, in opensuse-leap 16.0

ZDRVVX(3) 💅

Library Functions Manual · okay bestie, we’re checking the nonsymmetric eigenvalue expert driver and we are going to be so thorough about it 🫶

🎀 NAME 🎀

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

That’s the file this page is about. No cap, that’s the whole NAME situation. ✨

🌸 SYNOPSIS 🌸

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

🧚 Functions/Subroutines 🧚

  • 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 🦋

Here’s the full typed-out signature, because she is long and iconic:

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 🎀

ZDRVVX checks the nonsymmetric eigenvalue problem expert driver ZGEEVX.

It uses both test matrices generated randomly depending on data supplied in the calling sequence, as well as data read from an input file and including precomputed condition numbers to which it compares the ones it computes. 💖

When ZDRVVX is called, a number of matrix sizes (NSIZES) and a number of matrix types (NTYPES) 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) | 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).
  • (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.
  • (3) | |VR(i)| - 1 | / ulp and largest component real VR(i) denotes the i-th column of VR.
  • (4) | |VL(i)| - 1 | / ulp and largest component real VL(i) denotes the i-th column of VL.
  • (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.
  • (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.
  • (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.
  • (8) — formula:
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.

  • (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 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. NSIZES 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

okay bestie, we're in the parameter deep-dive now — hydrate and keep up 💖

  • NTYPES — INTEGER. The number of elements in DOTYPE. NTYPES must be at least zero. If it is zero, no randomly generated test matrices are tested, but 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 — 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 ZDRVVX to continue the same random number sequence.
  • THRESH — DOUBLE PRECISION. A test will count as ... if the ..., 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.
  • NIUNIT — INTEGER. The FORTRAN unit number for reading in the data file of problems to solve.
  • NOUNIT — INTEGER. The FORTRAN unit number for printing out error messages (e.g., if a routine returns INFO not equal to 0.)
  • A — COMPLEX*16 array, dimension (LDA, max(NN,12)). 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, 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.)
  • H — COMPLEX*16 array, dimension (LDA, max(NN,12)). Another copy of the test matrix A, modified by ZGEEVX.
  • W — COMPLEX*16 array, dimension (max(NN,12)). Contains the eigenvalues of A.
  • W1 — 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.
  • VL — COMPLEX*16 array, dimension (LDVL, max(NN,12)). VL holds the computed left eigenvectors.
  • LDVL — INTEGER. Leading dimension of VL. Must be at least max(1,max(NN,12)).
  • VR — COMPLEX*16 array, dimension (LDVR, max(NN,12)). VR holds the computed right eigenvectors.
  • LDVR — INTEGER. Leading dimension of VR. Must be at least max(1,max(NN,12)).
  • LRE — COMPLEX*16 array, dimension (LDLRE, max(NN,12)). LRE holds the computed right or left eigenvectors.
  • LDLRE — INTEGER. Leading dimension of LRE. Must be at least max(1,max(NN,12)).
  • RCONDV — DOUBLE PRECISION array, dimension (N). RCONDV holds the computed reciprocal condition numbers for eigenvectors.
  • RCNDV1 — DOUBLE PRECISION array, dimension (N). RCNDV1 holds more computed reciprocal condition numbers for eigenvectors.
  • RCDVIN — DOUBLE PRECISION array, dimension (N). When COMP = .TRUE. RCDVIN holds the precomputed reciprocal condition numbers for eigenvectors to be compared with RCONDV.
  • RCONDE — DOUBLE PRECISION array, dimension (N). RCONDE holds the computed reciprocal condition numbers for eigenvalues.
  • RCNDE1 — DOUBLE PRECISION array, dimension (N). RCNDE1 holds more computed reciprocal condition numbers for eigenvalues.
  • RCDEIN — DOUBLE PRECISION array, dimension (N). When COMP = .TRUE. RCDEIN holds the precomputed reciprocal condition numbers for eigenvalues to be compared with RCONDE.
  • SCALE — DOUBLE PRECISION array, dimension (N). Holds information describing balancing of matrix.
  • SCALE1 — DOUBLE PRECISION array, dimension (N). Holds information describing balancing of matrix.
  • WORK — COMPLEX*16 array, dimension (NWORK).
  • RESULT — 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.
  • NWORK — 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.
  • RWORK — DOUBLE PRECISION array, dimension (2*max(NN,12)).
  • INFO — INTEGER. If 0, then successful exit. If <0, then input parameter -INFO is incorrect. If >0, ZLATMR, CLATMS, CLATME or ZGET23 returned an error code, and INFO is its absolute value.

🧚 Some Local Variables and Parameters

  • 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.

The following four arrays decode JTYPE:

  • KTYPE(j) — The general type (1-10) for type j.
  • KMODE(j) — The MODE value to be passed to the matrix generator for type j.
  • 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). (0 means irrelevant.)

👑 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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