TESTING/EIG/sdrvev.f(3)

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🌸 TESTING/EIG/sdrvev.f(3) 🌸

From the Library Functions Manual, bestie — this is the whole page for TESTING/EIG/sdrvev.f, and we're getting into it. 💅

💖 NAME 💖

TESTING/EIG/sdrvev.f — yes, that's the file, and this page is its whole vibe.

🦋 SYNOPSIS 🦋

Functions/Subroutines

       subroutine sdrvev (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)
           SDRVEV

✨ Function/Subroutine Documentation ✨

Here's the full signature, copied exactly so you can clock every argument:

   subroutine sdrvev (integer nsizes, integer, dimension( * ) nn, integer
       ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed,
       real thresh, 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( 7 ) result, real, dimension( * ) work, integer nwork,
       integer, dimension( * ) iwork, integer info)
       SDRVEV

SDRVEV — that's the subroutine we're hyping up. 🫶

💫 Purpose 💫

Okay so SDRVEV checks the nonsymmetric eigenvalue problem driver SGEEV. When SDRVEV is called, you specify a number of matrix sizes and a number of matrix types. For each size 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.

(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 whether largest component real

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

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

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

(5) 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.

(6) 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.

(7) 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 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 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 clustered 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 clustered 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 clustered 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. It's the number of sizes of matrices to use. If it is zero, SDRVEV 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, SDRVEV 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 SDRVEV to continue the same random number sequence.

THRESH

THRESH is REAL. A test will count as failed if the error, 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 REAL 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

H is REAL array, dimension (LDA, max(NN)). Another copy of the test matrix A, modified by SGEEV.

WR

WR is REAL array, dimension (max(NN)).

WI

  • WI — a REAL array, dimension (max(NN)). Bestie, this holds the real and imaginary parts of the eigenvalues of A. On exit, WR + WI*i are the eigenvalues of the matrix in A. 💅
  • WR1 — a REAL array, dimension (max(NN)). That's it, that's the whole description, no cap. 🫶
  • WI1 — a REAL array, dimension (max(NN)). Same energy as WR, WI — these arrays contain the eigenvalues of A, but the ones computed when SGEEV only computes a partial eigendecomposition, i.e. not the eigenvalues and left and right eigenvectors. Lowkey an important distinction. ✨
  • VL — a REAL array, dimension (LDVL, max(NN)). VL holds the computed left eigenvectors. 💜
  • LDVL — an INTEGER. Leading dimension of VL. Must be at least max(1,max(NN)). Not optional, she checks. 🙈
  • VR — a REAL array, dimension (LDVR, max(NN)). VR holds the computed right eigenvectors. 🦋
  • LDVR — an INTEGER. Leading dimension of VR. Must be at least max(1,max(NN)). Same rules, different array. 🎀
  • LRE — a REAL array, dimension (LDLRE,max(NN)). LRE holds the computed right or left eigenvectors. Flexible icon. ✨
  • LDLRE — an INTEGER. Leading dimension of LRE. Must be at least max(1,max(NN)). Yes, again. It's giving "consistency". 👑
  • RESULT — a REAL array, dimension (7). These are the values computed by the seven tests described above. Heads up: the values are currently limited to 1/ulp, to avoid overflow. 🫶
  • WORK — a REAL array, dimension (NWORK). That's the whole vibe. 💀
  • NWORK — an INTEGER. The number of entries in WORK. This must be at least 5*NN(j)+2*NN(j)**2 for all j. Write that down, bestie, fr fr. 📝
  • IWORK — an INTEGER array, dimension (max(NN)). Short and sweet. 🍓
  • INFO — an INTEGER. If 0, then everything ran OK. 🎉 Otherwise:
    • -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.

And one more thing, super important: if SLATMR, SLATMS, SLATME or SGEEV returns an error code, the absolute value of it is returned. 😭

🧚 Some Local Variables and Parameters 🧚

Like, the supporting cast. The background dancers. Here's who's who:

  • ZERO, ONE — Real 0 and 1. The originals. 💅
  • 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. Yes, square roots. Of the previous four. 🌀

And 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.) Yes it says "Selectw", we don't fix typos in here, we respect the source. 💀

✨ Author ✨

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

Definition at line 402 of file sdrvev.f. 🎀

🌸 Author 🌸

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