SDRGES(3)

📦 version 3.12.0, in opensuse-leap 16.0

🎀 NAME 🎀

okay so bestie, this whole page lives in the Library Functions Manual and it's filed under sdrges.f(3) — and the "name" it's registered under is literally just the source file path, no cap:

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

that's it. that's the name. iconic of LAPACK honestly 💅

✨ SYNOPSIS ✨

Functions/Subroutines

here's the whole squad of arguments, and yeah it's a long one, don't panic:

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

🌸 Function/Subroutine Documentation 🌸

subroutine SDRGES

full glow-up version of the signature, types and dimensions and all, verbatim because we do not play with this 💖

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

💜 Purpose

so like, what does this thing even DO?? SDRGES is the vibe check for the nonsymmetric generalized eigenvalue (Schur form) problem driver SGGES. that's the whole job. it checks SGGES. periodt.

SGGES 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, and those 2x2 blocks correspond 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.

so w(j) = alpha(j)/beta(j) is a root of the characteristic equation, and that is not a drill bestie:

                det( A - w(j) B ) = 0

optionally it also reorders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal block of the Schur forms. cute of it 🎀

when SDRGES is called, a number of matrix types and a number of matrix sizes get 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 tests (5), (11) and (13), which do not do that, they simply will not be compared, iykyk.

  • (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 here are 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 here are 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.
ok and yes, the intro up there did mention a test (13) — it's one of the three that don't get compared to THRESH. it is simply not in the list. we move. 💀

🔥 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, the flop era of matrices honestly
  • (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, matching, obsessed
        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 ) — that's a diagonal matrix with those diagonal entries, bestie
  • (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 — INTEGER. the number of sizes of matrices to use. if it is zero, SDRGES does nothing, like it just will not be participating. NSIZES >= 0.
  • NN — INTEGER array, dimension (NSIZES). an array containing the sizes to be used for the matrices. zero values will be skipped, no explanation, they're just not invited. NN >= 0.
  • NTYPES — 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, and that one 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. highkey the "read the fine print" parameter.
  • ISEED —
and ISEED is exactly where part 1 hands the aux cable to part 2 🫶 the rest of the parameters are right there waiting, bestie, we're not done, keep reading.

Okay so ISEED is an 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 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

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 is REAL 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

LDA is INTEGER. The leading dimension of A, B, S, and T. It must be at least 1 and at least max( NN ). 📏

B

B is REAL 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 REAL array, dimension (LDA, max(NN)). The Schur form matrix computed from A by SGGES. On exit, S contains the Schur form matrix corresponding to the matrix in A. 🌸

T

T is REAL array, dimension (LDA, max(NN)). The upper triangular matrix computed from B by SGGES. 🔺

Q

Q is REAL array, dimension (LDQ, max(NN)). The (left) orthogonal matrix computed by SGGES. ⬅️

LDQ

LDQ is INTEGER. The leading dimension of Q and Z. It must be at least 1 and at least max( NN ). 📐

Z

Z is REAL array, dimension( LDQ, max(NN) ). The (right) orthogonal matrix computed by SGGES. ➡️

ALPHAR

ALPHAR is REAL array, dimension (max(NN)). 🍓

ALPHAI

ALPHAI is REAL array, dimension (max(NN)). 🍓

BETA

BETA is REAL array, dimension (max(NN)). The generalized eigenvalues of (A,B) computed by SGGES. ( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the k-th generalized eigenvalue of A and B. 👑

WORK

WORK is REAL array, dimension (LWORK). 🛠️

LWORK

LWORK is INTEGER. The dimension of the array WORK. LWORK >= MAX( 10*(N+1), 3*N*N ), where N is the largest matrix dimension. 🔢

RESULT

RESULT is REAL array, dimension (15). The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. ⚠️

BWORK

BWORK is LOGICAL array, dimension (N). 🧠

INFO

INFO is 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 sdrges.f.

Author

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

💖 That’s the whole page, bestie — LAPACK Version 3.12.0, source /home/abui...IG/sdrges.f(3), and the footer date is not shown in this extract, ngl 😭. Periodt. ✨

📦 where this page comes from