TESTING/EIG/cdrvev.f(3)

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version 3.12.1 · 3 versions
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hey bestie! 💖 welcome to the TESTING/EIG/cdrvev.f page from the Library Functions Manual, section 3. this is all about the cdrvev subroutine — it's lowkey iconic for checking the nonsymmetric eigenvalue problem driver CGEEV. let's get into it! ✨

💖 NAME 💖

TESTING/EIG/cdrvev.f — the file that brings you the cdrvev subroutine, no cap. 🫶

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine cdrvev (nsizes, nn, ntypes, dotype, iseed, thresh, nounit, a,
     lda, h, w, w1, vl, ldvl, vr, ldvr, lre, ldlre, result, work, nwork,
     rwork, iwork, info)
     CDRVEV

🦋 Function/Subroutine Documentation 🦋

subroutine cdrvev (integer nsizes, integer, dimension( * ) nn, integer
     ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed,
     real thresh, integer nounit, complex, dimension( lda, * ) a, integer
     lda, complex, dimension( lda, * ) h, complex, dimension( * ) w, complex,
     dimension( * ) w1, complex, dimension( ldvl, * ) vl, integer ldvl,
     complex, dimension( ldvr, * ) vr, integer ldvr, complex, dimension(
     ldlre, * ) lre, integer ldlre, real, dimension( 7 ) result, complex,
     dimension( * ) work, integer nwork, real, dimension( * ) rwork, integer,
     dimension( * ) iwork, integer info)
     CDRVEV

🌈 Purpose 🌈

CDRVEV checks the nonsymmetric eigenvalue problem driver CGEEV. 💅

When CDRVEV is called, a number of matrix sizes (NSIZES) and a number of matrix types (NTYPES) are specified. For each size (from NN) 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. Test (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. 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.
  3. Test (3): | |VR(i)| - 1 | / ulp and whether largest component real VR(i) denotes the i-th column of VR.
  4. Test (4): | |VL(i)| - 1 | / ulp and whether largest component real VL(i) denotes the i-th column of VL.
  5. Test (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. Test (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. Test (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 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.

🎀 Parameters 🎀

  • NSIZES (INTEGER) — the number of sizes of matrices to use. if it's zero, CDRVEV does nothing. it must be at least zero. 🫶
  • NN (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 (INTEGER) — the number of elements in DOTYPE. if it's zero, CDRVEV does nothing. it must be at least zero. 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. 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 CDRVEV to continue the same random number sequence.
  • THRESH (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 (INTEGER) — the FORTRAN unit number for printing out error messages (e.g., if a routine returns INFO not equal to 0.)
  • A (COMPLEX 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 (INTEGER) — the leading dimension of A, and H. LDA must be at least 1 and at least max(NN).
  • H (COMPLEX array, dimension (LDA, max(NN))) — another copy of the test matrix A, modified by CGEEV.
  • W (COMPLEX array, dimension (max(NN))) — the eigenvalues of A. on exit, W are the eigenvalues of the matrix in A.
  • W1 (COMPLEX array, dimension (max(NN))) — like W, this array contains the eigenvalues of A, but those computed when CGEEV only computes a partial eigendecomposition, i.e. not the eigenvalues and left and right eigenvectors.
  • VL — (description continues in part 2, bestie! 💖)

okay, still in the argument list — these next ones are the moment, bestie 💅

  • VL — a COMPLEX 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)). no skipping, bestie 💅
  • VR — a COMPLEX 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 — a COMPLEX 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 — a REAL array, dimension (7). The values computed by the seven tests described above. The values are currently limited to 1/ulp, to avoid overflow. ✨
  • WORK — a COMPLEX 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. 🫶
  • RWORK — a REAL array, dimension (2*max(NN)). 🌸
  • IWORK — an INTEGER array, dimension (max(NN)). 🌸
  • INFO — INTEGER. If 0, then everything ran OK. Otherwise, here's the whole lore:
    • -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) ).
    • -14: LDVL < 1 or LDVL < NMAX, where NMAX is max( NN(j) ).
    • -16: LDVR < 1 or LDVR < NMAX, where NMAX is max( NN(j) ).
    • -18: LDLRE < 1 or LDLRE < NMAX, where NMAX is max( NN(j) ).
    • -21: NWORK too small.
    • If CLATMR, CLATMS, CLATME or CGEEV 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 . 👑
  • 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 387 of file cdrvev.f.

🎀 Author 🎀

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that's the whole page, bestie — footer energy: LAPACK Version 3.12.0, page TESTING/EIG/cdrvev.f(3), and no date in the original footer so I'm not inventing one, periodt 💖✨

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