DDRVEV(3)

📦 version 3.12.0, in opensuse-leap 16.0

DDRVEV(3) 💖 Library Functions Manual 💖

okay bestie, this is part 1 and we’re going until the page literally cuts off at H. No sign-off, no goodbye — just full manual-page energy. 💅

🌸 NAME 🌸

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

That’s the file behind this whole DDRVEV(3) moment. Iconic path, honestly. 🎀

✨ SYNOPSIS ✨

🎀 Functions/Subroutines 🎀

  • subroutine DDRVEV (nsizes, nn, ntypes, dotype, iseed, thresh, nounit, a, lda, h, wr, wi, wr1, wi1, vl, ldvl, vr, ldvr, lre, ldlre, result, work, nwork, iwork, info) — DDRVEV
subroutine DDRVEV (nsizes, nn, ntypes, dotype, iseed, thresh, nounit, a,
    lda, h, wr, wi, wr1, wi1, vl, ldvl, vr, ldvr, lre, ldlre, result,
    work, nwork, iwork, info)
    DDRVEV

🦋 Function/Subroutine Documentation 🦋

🌟 subroutine DDRVEV 🌟

subroutine DDRVEV (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( lda, * ) h, double
     precision, dimension( * ) wr, double precision, dimension( * ) wi,
     double precision, dimension( * ) wr1, double precision, dimension( * )
     wi1, double precision, dimension( ldvl, * ) vl, integer ldvl, double
     precision, dimension( ldvr, * ) vr, integer ldvr, double precision,
     dimension( ldlre, * ) lre, integer ldlre, double precision, dimension( 7
     ) result, double precision, dimension( * ) work, integer nwork, integer,
     dimension( * ) iwork, integer info)
     DDRVEV

🎯 Purpose 🎯

quick vibe check: a few words are literally blank in the original text, so I’m marking those spots [blank] instead of guessing. no cap, we keep the facts exactly as they are. 😭

DDRVEV checks the nonsymmetric eigenvalue problem driver DGEEV.

When DDRVEV is called, a number of matrix [blank] () and a number of matrix [blank] are specified. For each size [blank] () and each type of matrix, one matrix will be generated and used to test the nonsymmetric eigenroutines. For each matrix, 7 tests will be performed:

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

Here VR is the matrix of unit right eigenvectors. W is a block diagonal matrix, with a 1x1 block for each real eigenvalue and a 2x2 block for each complex conjugate pair. If eigenvalues j and j+1 are a complex conjugate pair, so WR(j) = WR(j+1) = wr and WI(j) = - WI(j+1) = wi, then the 2 x 2 block corresponding to the pair will be:

(  wr  wi  )
( -wi  wr  )

Such a block multiplying an n x 2 matrix ( ur ui ) on the right will be the same as multiplying ur + i*ui by wr + i*wi.

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

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

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

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

  1. W(full) = W(partial)

W(full) denotes the eigenvalues computed when both VR and VL are also computed, and W(partial) denotes the eigenvalues computed when only W, only W and VR, or only W and VL are computed.

  1. VR(full) = VR(partial)

VR(full) denotes the right eigenvectors computed when both VR and VL are computed, and VR(partial) denotes the result when only VR is computed.

  1. VL(full) = VL(partial)

VL(full) denotes the left eigenvectors computed when both VR and VL are also computed, and VL(partial) denotes the result when only VL is computed.

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 (transposed) Jordan block, with 1's on the diagonal.
  • (4) A diagonal matrix with evenly spaced entries 1, ..., ULP and random signs. (ULP = (first number larger than 1) - 1 )
  • (5) A diagonal matrix with geometrically spaced entries 1, ..., ULP and random signs.
  • (6) A diagonal matrix with [blank] entries 1, ULP, ..., ULP and random signs.
  • (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 orthogonal and T has evenly spaced entries 1, ..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle.
  • (10) A matrix of the form U' T U, where U is orthogonal and T has geometrically spaced entries 1, ..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle.
  • (11) A matrix of the form U' T U, where U is orthogonal and T has [blank] entries 1, ULP,..., ULP with random signs on the diagonal and random O(1) entries in the upper triangle.
  • (12) A matrix of the form U' T U, where U is orthogonal and T has real or complex conjugate paired eigenvalues randomly chosen from ( ULP, 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 signs 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 signs 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 [blank] entries 1, ULP,..., ULP with random signs 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 real or complex conjugate paired eigenvalues randomly chosen from ( ULP, 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 (-1,1). 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.

🧸 Parameters 🧸

  • NSIZES — NSIZES is INTEGER. The number of sizes of matrices to use. If it is zero, DDRVEV does nothing. It must be at least zero.
  • 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. If it is zero, DDRVEV 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. .
  • DOTYPE — DOTYPE is 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 — ISEED is 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 DDRVEV to continue the same random number sequence.
  • THRESH — THRESH is DOUBLE PRECISION. A test will count as [blank] if the [blank], 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 — NOUNIT is INTEGER. The FORTRAN unit number for printing out error messages (e.g., if a routine returns INFO not equal to 0.)
  • A — A is 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.
  • LDA — LDA is INTEGER. The leading dimension of A, and H. LDA must be at least 1 and at least max(NN).
  • H — the original snippet stops right here at H, so the rest of this parameter’s description is coming in part 2. We’re not ending the page — just continuing the lore. 💅
  • H — DOUBLE PRECISION array, dimension (LDA, max(NN)). It’s another copy of the test matrix A, modified by DGEEV. 💅
  • WR — DOUBLE PRECISION array, dimension (max(NN)).
  • WI — DOUBLE PRECISION array, dimension (max(NN)).

The real and imaginary parts of the eigenvalues of A. On exit, WR + WI*i are the eigenvalues of the matrix in A.

  • WR1 — DOUBLE PRECISION array, dimension (max(NN)).
  • WI1 — DOUBLE PRECISION array, dimension (max(NN)).

Like WR, WI, these arrays contain the eigenvalues of A, but those computed when DGEEV only computes a partial eigendecomposition, i.e. not the eigenvalues and left and right eigenvectors.

  • VL — DOUBLE PRECISION array, dimension (LDVL, max(NN)). VL holds the computed left eigenvectors. 🦋
  • LDVL — INTEGER. Leading dimension of VL. Must be at least max(1,max(NN)).
  • VR — DOUBLE PRECISION array, dimension (LDVR, max(NN)). VR holds the computed right eigenvectors. ✨
  • LDVR — INTEGER. Leading dimension of VR. Must be at least max(1,max(NN)).
  • LRE — DOUBLE PRECISION array, dimension (LDLRE,max(NN)). LRE holds the computed right or left eigenvectors.
  • LDLRE — INTEGER. Leading dimension of LRE. Must be at least max(1,max(NN)).
  • RESULT — DOUBLE PRECISION array, dimension (7). The values computed by the seven tests described above. The values are currently limited to 1/ulp, to avoid overflow. 🔥
  • WORK — DOUBLE PRECISION array, dimension (NWORK).
  • NWORK — INTEGER. The number of entries in WORK. This must be at least 5*NN(j)+2*NN(j)**2 for all j.
  • IWORK — INTEGER array, dimension (max(NN)).
  • INFO — INTEGER. If 0, then everything ran OK.
    • -1: NSIZES < 0
    • -2: Some NN(j) < 0
    • -3: NTYPES < 0
    • -6: THRESH < 0
    • -9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ).
    • -16: LDVL < 1 or LDVL < NMAX, where NMAX is max( NN(j) ).
    • -18: LDVR < 1 or LDVR < NMAX, where NMAX is max( NN(j) ).
    • -20: LDLRE < 1 or LDLRE < NMAX, where NMAX is max( NN(j) ).
    • -23: NWORK too small.
    • If DLATMR, SLATMS, SLATME or DGEEV returns an error code, the absolute value of it is returned.

🧸 Some Local Variables and Parameters:

  • ZERO, ONE — real 0 and 1.
  • MAXTYP — the number of types defined.
  • NMAX — largest value in NN.
  • 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 402 of file ddrvev.f.

🌸 Author

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