DDRGES(3)

💖 DDRGES(3) 💖

From the Library Functions Manual, bestie — this is the DDRGES page, and we are doing the whole thing, no skips, no cap.

🌸 NAME 🌸

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

✨ 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 📚

`subroutine DDRGES`

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 sizes (NN) and a number of matrix types (DOTYPE) are specified. For each size in 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) | 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. This is the no-sorting-of-eigenvalues version.

(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. This is the with-sorting-of-eigenvalues version.

(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 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)

        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 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 )
          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, 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 — it's the seed for the random number generator: INTEGER array, dimension (4). On entry, ISEED specifies the seed. The array elements should be between 0 and 4095; if not, they get reduced mod 4096. Also, ISEED(4) must be odd. The random number generator uses a linear congruential sequence limited to small integers, so it 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 — a DOUBLE PRECISION threshold. A test will count as failed if the error ratio, 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 — an 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)). These are 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 exit status, bestie:
- = 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

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