/home/abui.../cdrges3.f(3)

💖 NAME 💖

/home/abui.../cdrges3.f(3) — Library Functions Manual. Bestie, this is the page for:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/TESTING/EIG/cdrges3.f

🌸 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 🦋

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 checks the nonsymmetric generalized eigenvalue (Schur form) problem driver CGGES3. It's literally the vibe check for that driver, no cap.

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

When CDRGES3 is called, a number of matrix sizes and a number of matrix types 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 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 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
(5)  ( J , J  )       (a pair of transposed Jordan blocks)

(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 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:

(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
(16) Q ( J , J ) Z     where Q and Z are random orthogonal matrices.

(17) — where T1 and T2 are upper triangular 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) — with s = machine precision.:

(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) —:

(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) —:

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

(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) —:

(22) Q ( big*T1, small*T2 ) Z    diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
                                 diag(T2) = ( 0, 1, ..., 1, 0, 0 )

(23) —:

(23) Q ( small*T1, big*T2 ) Z    diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
                                 diag(T2) = ( 0, 1, ..., 1, 0, 0 )

(24) —:

(24) Q ( small*T1, small*T2 ) Z  diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
                                 diag(T2) = ( 0, 1, ..., 1, 0, 0 )

(25) —:

(25) Q ( big*T1, big*T2 ) Z      diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
                                 diag(T2) = ( 0, 1, ..., 1, 0, 0 )

(26) — 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 — INTEGER. The number of sizes of matrices to use. If it is zero, SDRGES3 does nothing. NSIZES >= 0.
NN — INTEGER array, dimension (NSIZES). An array containing the sizes to be used for the matrices. Zero values will be skipped. NN >= 0.
NTYPES — 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 — 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 SDRGES3 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. 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 — (description continues in the next part, bestie — hold that thought 💅)

Argument roll call, continued — every single one, no skips, no cap. 💅

A — COMPLEX array, dimension(LDA, max(NN)). 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. It’s used as input only if 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)). Just the type/dimension tea here — no extra description in this block. 🍓
BETA — COMPLEX array, dimension (max(NN)). The generalized eigenvalues of (A,B) computed by CGGES3. And the iconic formula: 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. The exit-code fortune cookie:
- = 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

Generated automatically by Doxygen for LAPACK from the source code. ✨

That’s the page, bestie! 🎀 The footer energy is LAPACK Version 3.12.0 and the source path /home/abui.../cdrges3.f(3) — the footer doesn’t give a separate calendar date, just that version + file combo, so we’re not inventing one. 💖

📦 where this page comes from

opensuse-leap 16.0: lapack-man 3.12.0-160000.2.2 · BSD-3-Clause
opensuse-leap 16.0: lapack-man 3.12.0-160000.2.2 · BSD-3-Clause
opensuse-leap 16.0: lapack-man 3.12.0-160000.2.2 · BSD-3-Clause
opensuse-leap 16.0: lapack-man 3.12.0-160000.2.2 · BSD-3-Clause
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms