TESTING/EIG/cdrges3.f(3)

📦
version 3.12.1 · 2 versions
, in arch rolling

🌸 NAME 🌸

TESTING/EIG/cdrges3.f — okay bestie, this is the manual page for it, and the Library Functions Manual is literally right there in the header holding the vibe together 💅

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine cdrges3 (nsizes, nn, ntypes, dotype, iseed, thresh, nounit,
    a, lda, b, s, t, q, ldq, z, alpha, beta, work, lwork, rwork, result,
    bwork, info)

CDRGES3

🧚 Function/Subroutine Documentation 🧚

subroutine cdrges3 (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, * ) b, complex, dimension( lda, * ) s,
    complex, dimension( lda, * ) t, complex, dimension( ldq, * ) q, integer
    ldq, complex, dimension( ldq, * ) z, complex, dimension( * ) alpha,
    complex, dimension( * ) beta, complex, dimension( * ) work, integer
    lwork, real, dimension( * ) rwork, real, dimension( 13 ) result,
    logical, dimension( * ) bwork, integer info)

CDRGES3

💜 Purpose 💜

okay so CDRGES3 is the vibe check for the nonsymmetric generalized eigenvalue (Schur form) problem driver CGGES3.

CGGES3 factors A and B as Q*S*Z' and Q*T*Z', where ' means conjugate transpose, S and T are upper triangular (i.e., in generalized Schur form), and Q and Z are unitary. 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. main character rearrangement energy, honestly ✨

When CDRGES3 is called, a number of matrix sizes (NSIZES) and a number of matrix types (NTYPES) are specified. For each size (NN) 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. (1) — | A - Q S Z' | / ( |A| n ulp ) (no sorting of eigenvalues)
  2. (2) — | B - Q T Z' | / ( |B| n ulp ) (no sorting of eigenvalues)
  3. (3) — | I - QQ' | / ( n ulp ) (no sorting of eigenvalues)
  4. (4) — | I - ZZ' | / ( n ulp ) (no sorting of eigenvalues)
  5. (5) — if A is in Schur form (i.e. triangular form) (no sorting of eigenvalues)
  6. (6) — if eigenvalues = diagonal elements of the Schur form (S, T), i.e., test the maximum over j of D(j) where:
                    |alpha(j) - S(j,j)|        |beta(j) - T(j,j)|
          D(j) = ------------------------ + -----------------------
                 max(|alpha(j)|,|S(j,j)|)   max(|beta(j)|,|T(j,j)|)

(no sorting of eigenvalues)

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

(with sorting of eigenvalues).

  1. (12) — if sorting worked and SDIM is the number of eigenvalues which were CELECTed.
bestie, test (13) is named in that exception line but this page doesn't give its description — it goes straight into Test Matrices. No cap, that's just how the page runs. 💀

🌈 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 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) — ( 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) — ( J^t, J^t ) (a pair of transposed Jordan blocks)

        t   t
(5)  ( J , J  )

(6) — ( X, Y ) where:

                                     t                ( I   0  )
(6)  ( 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) — Q ( J^t, J^t ) Z where Q and Z are random orthogonal matrices.

           t   t
(16) 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 )

(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 with:

(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 with:

(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 with:

                                                        N-5
(20) Q ( T1, T2 ) Z    diag(T1)=( 0, 0, 1, 1, a, ..., a   =s, 0 )
                        diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )

(21) — Q ( T1, T2 ) Z with:

(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 with:

(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 with:

(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 with:

(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 with:

(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.

(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, SDRGES3 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, SDRGES3 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

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 SDRGES3 to continue the same random number sequence.

THRESH

THRESH is REAL — a test will count as a failure 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. THRESH >= 0.

NOUNIT

NOUNIT is INTEGER — the FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.)

A

A — the part-1 page cuts off here; part 2 picks up with A's description. 🫶

  • A — COMPLEX array, dimension(LDA, max(NN)). This one holds the original A matrix. It’s 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 — COMPLEX array, dimension(LDA, max(NN)). Holds the original B matrix. Input-only under the same exact conditions: NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. 🦋
  • S — COMPLEX array, dimension (LDA, max(NN)). The Schur form matrix computed from A by CGGES3. On exit, S contains the Schur form matrix corresponding to the matrix in A. 👑
  • T — COMPLEX array, dimension (LDA, max(NN)). The upper triangular matrix computed from B by CGGES3. 🍓
  • Q — COMPLEX array, dimension (LDQ, max(NN)). The (left) orthogonal matrix computed by CGGES3. 🫶
  • LDQ — INTEGER. The leading dimension of Q and Z. It must be at least 1 and at least max( NN ). 💜
  • Z — COMPLEX array, dimension( LDQ, max(NN) ). The (right) orthogonal matrix computed by CGGES3. 🌸
  • ALPHA — COMPLEX array, dimension (max(NN)). ✨
  • BETA — COMPLEX array, dimension (max(NN)). The generalized eigenvalues of (A,B) computed by CGGES3. ALPHA(k) / BETA(k) is the k-th generalized eigenvalue of A and B. 🔥
  • WORK — COMPLEX array, dimension (LWORK). 🧚
  • LWORK — INTEGER. The dimension of the array WORK. LWORK >= 3*N*N. 💀
  • RWORK — REAL array, dimension ( 8*N ). Real workspace. 💖
  • 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 378 of file cdrges3.f.

Author 🌈

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