TESTING/EIG/sdrvsx.f(3)

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🎀 TESTING/EIG/sdrvsx.f(3) — Library Functions Manual 🎀

💖 NAME 💖

TESTING/EIG/sdrvsx.f

✨ SYNOPSIS ✨

🌸 Functions/Subroutines 🌸

subroutine sdrvsx (nsizes, nn, ntypes, dotype, iseed, thresh, niunit,
    nounit, a, lda, h, ht, wr, wi, wrt, wit, wrtmp, witmp, vs, ldvs,
    vs1, result, work, lwork, iwork, bwork, info)

SDRVSX

🦋 Function/Subroutine Documentation 🦋

subroutine sdrvsx (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( lda, * )
    ht, real, dimension( * ) wr, real, dimension( * ) wi, real, dimension( *
    ) wrt, real, dimension( * ) wit, real, dimension( * ) wrtmp, real,
    dimension( * ) witmp, real, dimension( ldvs, * ) vs, integer ldvs, real,
    dimension( ldvs, * ) vs1, real, dimension( 17 ) result, real, dimension(
    * ) work, integer lwork, integer, dimension( * ) iwork, logical,
    dimension( * ) bwork, integer info)

SDRVSX

✨ Purpose ✨

okay bestie, gather round, because SDRVSX is lowkey the vibe check for the nonsymmetric eigenvalue (Schur form) problem expert driver SGEESX. no cap, this is the routine that says “bestie, let’s test this thing properly” 💅

SDRVSX 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. it’s giving “random chaos meets precomputed receipts” ✨

When SDRVSX is called, a number of matrix () and a number of matrix are specified. For each size () and each type of matrix, one matrix will be generated and used to test the nonsymmetric eigenroutines. For each matrix, 15 tests will be performed:

  1. Test (1): 0 if T is in Schur form, 1/ulp otherwise (no sorting of eigenvalues).
  2. Test (2): | A - VS T VS' | / ( n |A| ulp ). Here VS is the matrix of Schur eigenvectors, and T is in Schur form (no sorting of eigenvalues).
  3. Test (3): | I - VS VS' | / ( n ulp ) (no sorting of eigenvalues).
  4. Test (4): 0 if WR+sqrt(-1)*WI are eigenvalues of T, 1/ulp otherwise (no sorting of eigenvalues).
  5. Test (5): 0 if T(with VS) = T(without VS), 1/ulp otherwise (no sorting of eigenvalues).
  6. Test (6): 0 if eigenvalues(with VS) = eigenvalues(without VS), 1/ulp otherwise (no sorting of eigenvalues).
  7. Test (7): 0 if T is in Schur form, 1/ulp otherwise (with sorting of eigenvalues).
  8. Test (8): | A - VS T VS' | / ( n |A| ulp ). Here VS is the matrix of Schur eigenvectors, and T is in Schur form (with sorting of eigenvalues).
  9. Test (9): | I - VS VS' | / ( n ulp ) (with sorting of eigenvalues).
  10. Test (10): 0 if WR+sqrt(-1)*WI are eigenvalues of T, 1/ulp otherwise. If workspace sufficient, also compare WR, WI with and without reciprocal condition numbers (with sorting of eigenvalues).
  11. Test (11): 0 if T(with VS) = T(without VS), 1/ulp otherwise. If workspace sufficient, also compare T with and without reciprocal condition numbers (with sorting of eigenvalues).
  12. Test (12): 0 if eigenvalues(with VS) = eigenvalues(without VS), 1/ulp otherwise. If workspace sufficient, also compare VS with and without reciprocal condition numbers (with sorting of eigenvalues).
  13. Test (13): if sorting worked and SDIM is the number of eigenvalues which were SELECTed. If workspace sufficient, also compare SDIM with and without reciprocal condition numbers.
  14. Test (14): if RCONDE the same no matter if VS and/or RCONDV computed.
  15. Test (15): if RCONDV the same no matter if VS and/or RCONDE 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 eigenvalue average and right invariant subspace. For these matrices, in addition to tests (1) to (15) we will compute the following two tests:

Test (16): |RCONDE - RCDEIN| / cond(RCONDE)

RCONDE is the reciprocal average eigenvalue condition number computed by SGEESX 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.

Test (17): |RCONDV - RCDVIN| / cond(RCONDV)

RCONDV is the reciprocal right invariant subspace condition number computed by SGEESX 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.

🎀 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 — 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. NTYPES must be at least zero. If it is 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
  • DOTYPE — okay so DOTYPE is a 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 SDRVSX to continue the same random number sequence. 🎲
  • THRESH — THRESH is REAL. A test will count as [blank in the original] if the [blank in the original] 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 — NIUNIT is INTEGER. The FORTRAN unit number for reading in the data file of problems to solve. 📥
  • 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 SGEESX. 🌀
  • HT — HT is REAL array, dimension (LDA, max(NN)). Yet another copy of the test matrix A, modified by SGEESX. 🌪️
  • WR — WR is REAL array, dimension (max(NN)). 🍓
  • WI — WI is REAL array, dimension (max(NN)). Together with WR, these are the real and imaginary parts of the eigenvalues of A. On exit, WR + WI*i are the eigenvalues of the matrix in A. 💫
  • WRT — WRT is REAL array, dimension (max(NN)). 🎀
  • WIT — WIT is REAL array, dimension (max(NN)). Like WR, WI, these arrays contain the eigenvalues of A, but those computed when SGEESX only computes a partial eigendecomposition, i.e. not Schur vectors. 🦋
  • WRTMP — WRTMP is REAL array, dimension (max(NN)). 🌸
  • WITMP — WITMP is REAL array, dimension (max(NN)). More temporary storage for eigenvalues. 🧺
  • VS — VS is REAL array, dimension (LDVS, max(NN)). VS holds the computed Schur vectors. 👑
  • LDVS — LDVS is INTEGER. Leading dimension of VS. Must be at least max(1,max(NN)). 📐
  • VS1 — VS1 is REAL array, dimension (LDVS, max(NN)). VS1 holds another copy of the computed Schur vectors. 🪞
  • RESULT — RESULT is REAL array, dimension (17). The values computed by the 17 tests described above. The values are currently limited to 1/ulp, to avoid overflow. 🔥
  • WORK — WORK is REAL array, dimension (LWORK). 🛠️
  • LWORK — LWORK is INTEGER. The number of entries in WORK. This must be at least max(3*NN(j),2*NN(j)**2) for all j. 🧮
  • IWORK — IWORK is INTEGER array, dimension (max(NN)*max(NN)). 🧱
  • BWORK — BWORK is LOGICAL array, dimension (max(NN)). 🪄
  • INFO — INFO is INTEGER. If 0, successful exit.
    • <0, input parameter -INFO is incorrect.
    • >0, SLATMR, SLATMS, SLATME or SGET24 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.
  • 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 [blank].
  • KMODE(j) — The MODE value to be passed to the matrix generator for type [blank].
  • 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 450 of file sdrvsx.f.

Author 🤖

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