SDRVVX(3)

📦 version 3.12.0, in opensuse-leap 16.0

okay bestie, this is SDRVVX(3) from the Library Functions Manual—and the header is literally /home/abui...IG/sdrvvx.f(3) Library Functions Manual/home/abui...IG/sdrvvx.f(3). We are keeping every single detail exactly where it belongs, just with the group-chat energy turned all the way up. 💅✨

🌸 NAME 🌸

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

🦋 SYNOPSIS 🦋

Functions/Subroutines

Here’s the call, character for character:

subroutine SDRVVX (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)

SDRVVX

💖 Function/Subroutine Documentation 💖

Okay so here’s the typed-out signature—same order, same dimensions, same everything:

subroutine SDRVVX (integer nsizes, integer, dimension( * ) nn, integer
       ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed,
       real thresh, integer niunit, integer nounit, real, dimension( lda, * )
       a, integer lda, real, dimension( lda, * ) h, real, dimension( * ) wr,
       real, dimension( * ) wi, real, dimension( * ) wr1, real, dimension( * )
       wi1, real, dimension( ldvl, * ) vl, integer ldvl, real, dimension( ldvr,
       * ) vr, integer ldvr, real, dimension( ldlre, * ) lre, integer ldlre,
       real, dimension( * ) rcondv, real, dimension( * ) rcndv1, real,
       dimension( * ) rcdvin, real, dimension( * ) rconde, real, dimension( * )
       rcnde1, real, dimension( * ) rcdein, real, dimension( * ) scale, real,
       dimension( * ) scale1, real, dimension( 11 ) result, real, dimension( *
       ) work, integer nwork, integer, dimension( * ) iwork, integer info)

SDRVVX

🎯 Purpose 🎯

okay so SDRVVX is the vibe check for the nonsymmetric eigenvalue problem expert driver SGEEVX. It’s not doing the math for fun—it’s testing it.

SDRVVX 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. Yes, both. She’s thorough. 💅

When SDRVVX is called, a number of matrix () and a number of matrix 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. Test (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. Test (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. Test (3):
| |VR(i)| - 1 | / ulp and largest component real

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

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

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

  1. Test (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. Test (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. Test (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. Test (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. Test (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 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. Test (10):
|RCONDV - RCDVIN| / cond(RCONDV)

RCONDV is the reciprocal right eigenvector condition number computed by SGEEVX 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. Test (11):
|RCONDE - RCDEIN| / cond(RCONDE)

RCONDE is the reciprocal eigenvalue condition number computed by SGEEVX 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. It’s 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 so picking right back up with the argument tea, no cap:

  • NTYPES — INTEGER. The number of elements in DOTYPE. NTYPES must be at least zero. If it’s zero, no randomly generated test matrices are tested, but and 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 SDRVVX to continue the same random number sequence.
  • THRESH — REAL. 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 — REAL 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 — REAL array, dimension (LDA, max(NN,12)). Another copy of the test matrix A, modified by SGEEVX.
  • WR — REAL array, dimension (max(NN)).
  • WI — REAL 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 — REAL array, dimension (max(NN,12)).
  • WI1 — REAL array, dimension (max(NN,12)). Like WR, WI, these arrays contain the eigenvalues of A, but those computed when SGEEVX only computes a partial eigendecomposition, i.e. not the eigenvalues and left and right eigenvectors.
  • VL — REAL 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 — REAL 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 — REAL 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 — REAL array, dimension (N). RCONDV holds the computed reciprocal condition numbers for eigenvectors.
  • RCNDV1 — REAL array, dimension (N). RCNDV1 holds more computed reciprocal condition numbers for eigenvectors.
  • RCDVIN — REAL array, dimension (N). When COMP = .TRUE. RCDVIN holds the precomputed reciprocal condition numbers for eigenvectors to be compared with RCONDV.
  • RCONDE — REAL array, dimension (N). RCONDE holds the computed reciprocal condition numbers for eigenvalues.
  • RCNDE1 — REAL array, dimension (N). RCNDE1 holds more computed reciprocal condition numbers for eigenvalues.
  • RCDEIN — REAL array, dimension (N). When COMP = .TRUE. RCDEIN holds the precomputed reciprocal condition numbers for eigenvalues to be compared with RCONDE.
  • SCALE — REAL array, dimension (N). Holds information describing balancing of matrix.
  • SCALE1 — REAL array, dimension (N). Holds information describing balancing of matrix.
  • RESULT — REAL array, dimension (11). The values computed by the seven tests described above. The values are currently limited to 1/ulp, to avoid overflow.
  • WORK — REAL 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. If 0, then successful exit. If <0, then input parameter -INFO is incorrect. If >0, SLATMR, SLATMS, SLATME or SGET23 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 .
  • 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.)

👑 Author ✨

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

Definition at line 515 of file sdrvvx.f.

🤖 Author — auto-generated edition ✨

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