TESTING/EIG/ddrges.f(3)

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🎀 TESTING/EIG/ddrges.f — Library Functions Manual 🎀

💖 NAME 💖

TESTING/EIG/ddrges.f — that's the name, bestie. Periodt. ✨

🦋 SYNOPSIS 🦋

Functions/Subroutines

       subroutine ddrges (nsizes, nn, ntypes, dotype, iseed, thresh, nounit, a,
           lda, b, s, t, q, ldq, z, alphar, alphai, beta, work, lwork, result,
           bwork, info)
           DDRGES

✨ Function/Subroutine Documentation ✨

okay so this is the actual subroutine, no cap:

   subroutine ddrges (integer nsizes, integer, dimension( * ) nn, integer
       ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed,
       double precision thresh, integer nounit, double precision, dimension(
       lda, * ) a, integer lda, double precision, dimension( lda, * ) b, double
       precision, dimension( lda, * ) s, double precision, dimension( lda, * )
       t, double precision, dimension( ldq, * ) q, integer ldq, double
       precision, dimension( ldq, * ) z, double precision, dimension( * )
       alphar, double precision, dimension( * ) alphai, double precision,
       dimension( * ) beta, double precision, dimension( * ) work, integer
       lwork, double precision, dimension( 13 ) result, logical, dimension( * )
       bwork, integer info)
       DDRGES

Purpose: 🫶

okay so DDRGES checks the nonsymmetric generalized eigenvalue (Schur form) problem driver DGGES. 💅

DGGES 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 DDRGES 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
(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 D1 is diag( 0, 0, 1, ..., N-3, 0 ) and D2 is diag( 0, N-3, N-4,..., 1, 0, 0 )
(16) — Q ( J , J ) 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 )
(18) — Q ( T1, T2 ) Z with diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 ), diag(T2) = ( 0, 1, 0, 1,..., 1, 0 ), and s = machine precision.
(19) — Q ( T1, T2 ) Z with diag(T1)=( 0,0,1,1, 1-d, ..., 1-(N-5)*d=s, 0 ) and 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 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 diag(T1) = ( 0, 0, 1, ..., N-3, 0 ) and diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(23) — Q ( small*T1, big*T2 ) Z with diag(T1) = ( 0, 0, 1, ..., N-3, 0 ) and diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(24) — Q ( small*T1, small*T2 ) Z with diag(T1) = ( 0, 0, 1, ..., N-3, 0 ) and diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(25) — Q ( big*T1, big*T2 ) Z with diag(T1) = ( 0, 0, 1, ..., N-3, 0 ) and 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, DDRGES 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, DDRGES 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 — 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 DDRGES to continue the same random number sequence. 💅
THRESH — DOUBLE PRECISION. 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 — INTEGER. The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.) 🫶
A — DOUBLE PRECISION 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 — DOUBLE PRECISION 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 — DOUBLE PRECISION array, dimension (LDA, max(NN)). The Schur form matrix computed from A by DGGES. On exit, S contains the Schur form matrix corresponding to the matrix in A. ⭐
T — DOUBLE PRECISION array, dimension (LDA, max(NN)). The upper triangular matrix computed from B by DGGES. 🍓
Q — DOUBLE PRECISION array, dimension (LDQ, max(NN)). The (left) orthogonal matrix computed by DGGES. 🔥
LDQ — INTEGER. The leading dimension of Q and Z. It must be at least 1 and at least max( NN ). 💜
Z — DOUBLE PRECISION array, dimension ( LDQ, max(NN) ). The (right) orthogonal matrix computed by DGGES. 🌈
ALPHAR — DOUBLE PRECISION array, dimension (max(NN)). 🎀
ALPHAI — DOUBLE PRECISION array, dimension (max(NN)). 💖
BETA — DOUBLE PRECISION array, dimension (max(NN)). The generalized eigenvalues of (A,B) computed by DGGES. ( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the k-th generalized eigenvalue of A and B. 👑
WORK — DOUBLE PRECISION 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 — DOUBLE PRECISION 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 ddrges.f. 📝

🪄 Author 🪄

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

💖 That’s the whole page, bestie! This one’s LAPACK Version 3.12.0, from TESTING/EIG/ddrges.f(3), with the footer date doing its timestamp moment — periodt. Go off and test those generalized Schur vibes ✨🎀

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