TESTING/EIG/cdrges.f(3)

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💅 TESTING/EIG/cdrges.f 💅

okay so this is from the Library Functions Manual — the TESTING/EIG/cdrges.f page, and it is a lot in the best way. grab your iced coffee, bestie, we're checking CDRGES 💖

🌸 NAME 🌸

TESTING/EIG/cdrges.f

✨ SYNOPSIS ✨

🦋 Functions/Subroutines 🦋

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

🎀 Function/Subroutine Documentation 🎀

here's the full signature, no cap:

subroutine cdrges (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)
    CDRGES

💖 Purpose 💖

okay so CDRGES checks the nonsymmetric generalized eigenvalue (Schur form) problem driver CGGES.

CGGES 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.

When CDRGES 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. triangular form) (no sorting of
      eigenvalues)

(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)

(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 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).

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

🧪 Test Matrices 🧪

the deets on the test matrices are literally this, bestie:

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)

        t   t
(5)  ( J , J  )       (a pair of transposed Jordan blocks)

                                    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  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 )
          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 )

(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 )

                                                       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    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 — 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 SDRGES to continue the same random number sequence.
THRESH — THRESH is REAL. A test will count as if the , 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 — okay so this is a COMPLEX 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 ). respect the math 🫶
B — this one is a COMPLEX 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 — S is a COMPLEX array, dimension (LDA, max(NN)). The Schur form matrix computed from A by CGGES. On exit, S contains the Schur form matrix corresponding to the matrix in A. iconic behavior 🌸
T — T is a COMPLEX array, dimension (LDA, max(NN)). The upper triangular matrix computed from B by CGGES. 🔥
Q — Q is a COMPLEX array, dimension (LDQ, max(NN)). The (left) orthogonal matrix computed by CGGES. 👑
LDQ — INTEGER. The leading dimension of Q and Z. It must be at least 1 and at least max( NN ). ⭐
Z — Z is a COMPLEX array, dimension ( LDQ, max(NN) ). The (right) orthogonal matrix computed by CGGES. 🦋
ALPHA — ALPHA is a COMPLEX array, dimension (max(NN)). Tiny but important, bestie 💖
BETA — BETA is a COMPLEX array, dimension (max(NN)). The generalized eigenvalues of (A,B) computed by CGGES. ALPHA(k) / BETA(k) is the k-th generalized eigenvalue of A and B. 🎀
WORK — WORK is a COMPLEX array, dimension (LWORK). ✨
LWORK — INTEGER. The dimension of the array WORK. LWORK >= 3*N*N. do not skip this check 😭
RWORK — RWORK is a REAL array, dimension ( 8*N ). Real workspace. 💜
RESULT — RESULT is a REAL array, dimension (15). The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. safety first 🫶
BWORK — BWORK is a LOGICAL array, dimension (N). 🍓
INFO — INTEGER. The whole status tea:
- = 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 cdrges.f.

🎀 Author 🎀

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Signed off with LAPACK Version 3.12.0, page TESTING/EIG/cdrges.f(3) — and the footer didn’t hand us a date, so I’m not inventing one, bestie 🙈💖

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