TESTING/EIG/sdrges.f(3)

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

okay bestie, this is the TESTING/EIG/sdrges.f page, and it is literally here to vibe-check the nonsymmetric generalized eigenvalue Schur form driver. 💅

🌸 NAME

TESTING/EIG/sdrges.f

✨ SYNOPSIS

Functions/Subroutines

  • subroutine sdrges (...) — SDRGES
subroutine sdrges (nsizes, nn, ntypes, dotype, iseed, thresh, nounit, a,
    lda, b, s, t, q, ldq, z, alphar, alphai, beta, work, lwork, result,
    bwork, info)

SDRGES — the icon, the moment, the subroutine. 👑

🛠️ Function/Subroutine Documentation

subroutine sdrges 🔥

subroutine sdrges (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, * ) b, real, dimension( lda, * ) s, real,
    dimension( lda, * ) t, real, dimension( ldq, * ) q, integer ldq, real,
    dimension( ldq, * ) z, real, dimension( * ) alphar, real, dimension( * )
    alphai, real, dimension( * ) beta, real, dimension( * ) work, integer
    lwork, real, dimension( 13 ) result, logical, dimension( * ) bwork,
    integer info)

SDRGES — that's the name, bestie. No cap. 💖

🎯 Purpose

Okay so SDRGES checks the nonsymmetric generalized eigenvalue (Schur form) problem driver SGGES.

SGGES factors A and B as Q S Z' and Q T Z', where ' means transpose, T is upper triangular, S is in generalized Schur form (block upper triangular, with 1x1 and 2x2 blocks on the diagonal, the 2x2 blocks corresponding to complex conjugate pairs of generalized eigenvalues), and Q and Z are orthogonal. It also computes the generalized eigenvalues (alpha(j),beta(j)), j=1,...,n. Thus, w(j) = alpha(j)/beta(j) is a root of the characteristic equation

det( A - w(j) B ) = 0

Optionally it also reorder the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal block of the Schur forms.

When SDRGES is called, a number of matrix () and a number of matrix are specified. For each size () and each TYPE of matrix, a pair of matrices (A, B) will be generated and used for testing. For each matrix pair, the following 13 tests will be performed and compared with the threshold THRESH except the tests (5), (11) and (13).

  • (1) — | A - Q S Z' | / ( |A| n ulp ) (no sorting of eigenvalues)
  • (2) — | B - Q T Z' | / ( |B| n ulp ) (no sorting of eigenvalues)
  • (3) — | I - QQ' | / ( n ulp ) (no sorting of eigenvalues)
  • (4) — | I - ZZ' | / ( n ulp ) (no sorting of eigenvalues)
  • (5) — if A is in Schur form (i.e. quasi-triangular form) (no sorting of eigenvalues)
  • (6) — if eigenvalues = diagonal blocks of the Schur form (S, T), i.e., test the maximum over j of D(j) where:
if alpha(j) is real:
              |alpha(j) - S(j,j)|        |beta(j) - T(j,j)|
    D(j) = ------------------------ + -----------------------
           max(|alpha(j)|,|S(j,j)|)   max(|beta(j)|,|T(j,j)|)

if alpha(j) is complex:
                          | det( s S - w T ) |
    D(j) = ---------------------------------------------------
           ulp max( s norm(S), |w| norm(T) )*norm( s S - w T )

and S and T are here the 2 x 2 diagonal blocks of S and T corresponding to the j-th and j+1-th eigenvalues. (no sorting of eigenvalues)

  • (7) — | (A,B) - Q (S,T) Z' | / ( | (A,B) | n ulp ) (with sorting of eigenvalues).
  • (8) — | I - QQ' | / ( n ulp ) (with sorting of eigenvalues).
  • (9) — | I - ZZ' | / ( n ulp ) (with sorting of eigenvalues).
  • (10) — if A is in Schur form (i.e. quasi-triangular form) (with sorting of eigenvalues).
  • (11) — if eigenvalues = diagonal blocks of the Schur form (S, T), i.e. test the maximum over j of D(j) where:
if alpha(j) is real:
              |alpha(j) - S(j,j)|        |beta(j) - T(j,j)|
    D(j) = ------------------------ + -----------------------
           max(|alpha(j)|,|S(j,j)|)   max(|beta(j)|,|T(j,j)|)

if alpha(j) is complex:
                          | det( s S - w T ) |
    D(j) = ---------------------------------------------------
           ulp max( s norm(S), |w| norm(T) )*norm( s S - w T )

and S and T are here the 2 x 2 diagonal blocks of S and T corresponding to the j-th and j+1-th eigenvalues. (with sorting of eigenvalues).

  • (12) — if sorting worked and SDIM is the number of eigenvalues which were SELECTed.

🎲 Test Matrices

The sizes of the test matrices 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) — ( 0, 0 ) — a pair of zero matrices.
  • (2) — ( I, 0 ) — an identity and a zero matrix.
  • (3) — ( 0, I ) — an identity and a zero matrix.
  • (4) — ( I, I ) — a pair of identity matrices.
  • (5) — a pair of transposed Jordan blocks:
     t   t
( J , J  )       (a pair of transposed Jordan blocks)
  • (6) — where the setup is:
                                    t                ( I   0  )
( X, Y )         where  X = ( J   0  )  and Y = (      t )
                             ( 0   I  )          ( 0   J  )
                      and I is a k x k identity and J a (k+1)x(k+1)
                      Jordan block; k=(N-1)/2
  • (7) — ( D, I ) where D is diag( 0, 1,..., N-1 ) (a diagonal matrix with those diagonal entries.)
  • (8) — ( I, D )
  • (9) — ( big*D, small*I ) where big is near overflow and small=1/big
  • (10) — ( small*D, big*I )
  • (11) — ( big*I, small*D )
  • (12) — ( small*I, big*D )
  • (13) — ( big*D, big*I )
  • (14) — ( small*D, small*I )
  • (15) — ( D1, D2 ) where D1 is diag( 0, 0, 1, ..., N-3, 0 ) and D2 is diag( 0, N-3, N-4,..., 1, 0, 0 )
  • (16) — where Q and Z are random orthogonal matrices:
           t   t
Q ( J , J ) Z     where Q and Z are random orthogonal matrices.
  • (17) — Q ( T1, T2 ) Z where T1 and T2 are upper triangular matrices with random O(1) entries above the diagonal and diagonal entries diag(T1) = ( 0, 0, 1, ..., N-3, 0 ) and diag(T2) = ( 0, N-3, N-4,..., 1, 0, 0 )
  • (18) — Q ( T1, T2 ) Z
diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 )
diag(T2) = ( 0, 1, 0, 1,..., 1, 0 )
s = machine precision.
  • (19) — Q ( T1, T2 ) Z
diag(T1)=( 0,0,1,1, 1-d, ..., 1-(N-5)*d=s, 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 )
  • (20) — Q ( T1, T2 ) Z
                                                        N-5
diag(T1)=( 0, 0, 1, 1, a, ..., a   =s, 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
  • (21) — Q ( T1, T2 ) Z
diag(T1)=( 0, 0, 1, r1, r2, ..., r(N-4), 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )

where r1,..., r(N-4) are random.

  • (22) — Q ( big*T1, small*T2 ) Z
diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
  • (23) — Q ( small*T1, big*T2 ) Z
diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
  • (24) — Q ( small*T1, small*T2 ) Z
diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
  • (25) — Q ( big*T1, big*T2 ) Z
diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
  • (26) — Q ( T1, T2 ) Z where T1 and T2 are random upper-triangular matrices.

🧾 Parameters

NSIZES

NSIZES is INTEGER. The number of sizes of matrices to use. If it is zero, SDRGES does nothing. NSIZES >= 0.

NN

NN is INTEGER array, dimension (NSIZES). An array containing the sizes to be used for the matrices. Zero values will be skipped. NN >= 0.

NTYPES

NTYPES is INTEGER. The number of elements in DOTYPE. If it is zero, SDRGES 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 on input. 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

Arguments, part 2 💅 (continuing the docs, no cap)

  • ISEED — INTEGER array, dimension (4). On entry, ISEED is literally the seed of the random number generator. The array elements should be between 0 and 4095; if they’re not, they will be reduced mod 4096. Also, ISEED(4) must be odd — periodt, required. 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 SDRGES to continue the same random number sequence. 🎲
  • THRESH — REAL. A test will count as the thing if the error measure, 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. THRESH >= 0. 🚨
  • NOUNIT — INTEGER. The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0). 📠
  • A — REAL array, dimension(LDA, max(NN)). Used to hold the original A matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE.
  • LDA — INTEGER. The leading dimension of A, B, S, and T. It must be at least 1 and at least max( NN ).
  • B — REAL array, dimension(LDA, max(NN)). Used to hold the original B matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE.
  • S — REAL array, dimension (LDA, max(NN)). The Schur form matrix computed from A by SGGES. On exit, S contains the Schur form matrix corresponding to the matrix in A. ✨
  • T — REAL array, dimension (LDA, max(NN)). The upper triangular matrix computed from B by SGGES.
  • Q — REAL array, dimension (LDQ, max(NN)). The (left) orthogonal matrix computed by SGGES.
  • LDQ — INTEGER. The leading dimension of Q and Z. It must be at least 1 and at least max( NN ).
  • Z — REAL array, dimension( LDQ, max(NN) ). The (right) orthogonal matrix computed by SGGES.
  • ALPHAR — REAL array, dimension (max(NN)).
  • ALPHAI — REAL array, dimension (max(NN)).
  • BETA — REAL array, dimension (max(NN)). The generalized eigenvalues of (A,B) computed by SGGES. ( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the k-th generalized eigenvalue of A and B. 🧮
  • WORK — REAL array, dimension (LWORK).
  • LWORK — INTEGER. The dimension of the array WORK. LWORK >= MAX( 10*(N+1), 3*N*N ), where N is the largest matrix dimension.
  • RESULT — REAL array, dimension (15). The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. 🔥
  • BWORK — LOGICAL array, dimension (N).
  • INFO — INTEGER. = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value.; > 0: A routine returned an error code. INFO is the absolute value of the INFO value returned. 💀

Author 🌸

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

Definition at line 399 of file sdrges.f.

Author (the Doxygen one) 🤖

Generated automatically by Doxygen for LAPACK from the source code.

Footer cuteness, bestie: LAPACK Version 3.12.0, page TESTING/EIG/sdrges.f(3) — and the date slot is not giving date in this snippet, so that’s the vibe. 🎀

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