DDRVVX(3)

📦 version 3.12.0, in opensuse-leap 16.0

💖 DDRVVX(3) 💖

okay so this is DDRVVX(3) from the Library Functions Manual, and the file path situation is /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/TESTING/EIG/ddrvvx.f. The header line also has /home/abui...IG/ddrvvx.f(3) on both sides, which is giving truncated-path realness, but we move.

✨ NAME ✨

The NAME section is literally that path: /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/TESTING/EIG/ddrvvx.f. Iconic, minimal, no notes.

🦋 SYNOPSIS 🦋

Functions/Subroutines

Here’s the subroutine lineup, bestie:

subroutine DDRVVX (nsizes, nn, ntypes, dotype, iseed, thresh, niunit,
    nounit, a, lda, h, wr, wi, wr1, wi1, vl, ldvl, vr, ldvr, lre, ldlre,
    rcondv, rcndv1, rcdvin, rconde, rcnde1, rcdein, scale, scale1,
    result, work, nwork, iwork, info)
    DDRVVX

and yes, that is a lot of arguments. The parenthesis budget is fully spent.

🎀 Function/Subroutine Documentation 🎀

subroutine DDRVVX — the whole typed-out signature

subroutine DDRVVX (integer nsizes, integer, dimension( * ) nn, integer
    ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed,
    double precision thresh, integer niunit, 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( * ) 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, double precision, dimension( * ) work, integer
    nwork, integer, dimension( * ) iwork, integer info)
    DDRVVX

🌈 Purpose 🌈

This icon DDRVVX checks the nonsymmetric eigenvalue problem expert driver DGEEVX.

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

When DDRVVX is called, a number of matrix () and a number of matrix are specified in the calling sequence (yes, the source has blanks there, bestie). 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 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 largest component real

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

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

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

  1. 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. 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. 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.

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. 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 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 (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 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 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 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

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:

  1. |RCONDV - RCDVIN| / cond(RCONDV)

RCONDV is the reciprocal right eigenvector condition number computed by DGEEVX 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.

  1. |RCONDE - RCDEIN| / cond(RCONDE)

RCONDE is the reciprocal eigenvalue condition number computed by DGEEVX 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

…and this is where part 1 stops, right at NN, mid-parameter, no cap.

  • NN — INTEGER array, dimension (NSIZES). This is literally just your list of sizes for the matrices, bestie 🎀. Zero values get skipped (they're not invited to the party), and every value has to be at least zero. Negative sizes? Not a vibe, not allowed.
  • NTYPES — INTEGER, the number of elements in DOTYPE. It must be at least zero. And here's the tea: if it's zero, no randomly generated test matrices get tested, but and test matrices read from NIUNIT will be tested ☕. 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. That trick 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. And if NTYPES is larger than MAXTYP, then DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored 🙈 (extra entries, no thoughts, head empty).
  • 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 — non-negotiable, that's the law 💅. 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 DDRVVX to continue the same random number sequence. So it's a little continuation token, iconic honestly 👑.
  • THRESH — DOUBLE PRECISION. A test will count as failed if the error ratio, computed as described above, exceeds THRESH. (The original doc line is a little haunted, ngl — it literally reads A test will count as if the , computed as described above, exceeds THRESH. — but you get the picture 🫶.) 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 — DOUBLE PRECISION 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 the arrays A and H. LDA >= max(NN,12), since 12 is the dimension of the largest matrix in the precomputed input file.
  • H — DOUBLE PRECISION array, dimension (LDA, max(NN,12)). Another copy of the test matrix A, modified by DGEEVX. Basically the glow-up version that got touched 😭.
  • WR — DOUBLE PRECISION array, dimension (max(NN)).
  • WI — DOUBLE PRECISION array, dimension (max(NN)). WR and WI hold 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,12)).
  • WI1 — DOUBLE PRECISION array, dimension (max(NN,12)). Like WR, WI, these arrays contain the eigenvalues of A, but those computed when DGEEVX only computes a partial eigendecomposition, i.e. not the eigenvalues and left and right eigenvectors 🦋.
  • VL — DOUBLE PRECISION 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 — DOUBLE PRECISION 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 — DOUBLE PRECISION 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 (yes, more — she's thorough 💅).
  • 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. (Same energy, second copy 🫶.)
  • 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 🔥.
  • WORK — DOUBLE PRECISION array, dimension (NWORK).
  • 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.
  • IWORK — INTEGER array, dimension (2*max(NN,12)).
  • INFO — INTEGER. The whole mood report, honestly:
    • If 0, then successful exit 🎉.
    • If <0, then input parameter -INFO is incorrect 😬.
    • If >0, DLATMR, SLATMS, SLATME or DGET23 returned an error code, and INFO is its absolute value 💀.

🧚 Some Local Variables and Parameters 🧚

the little guys behind the scenes

  • 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, iykyk 👑:

  • 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 516 of file ddrvvx.f.

🫶 Author (yes, again, don't come for me) 🫶

Generated automatically by Doxygen for LAPACK from the source code 🤖✨.

and that's the whole page, bestie — every last array, every last INFO value, all of it 🎀 no cap

Okay so that's the wrap 💖 this was LAPACK Version 3.12.0, straight out of /home/abui...IG/ddrvvx.f(3) — go forth and test those eigen-things, you absolute main character 🌈👑💅

📦 where this page comes from