/home/abui.../sdrgev3.f(3)

📦 version 3.12.0, in opensuse-leap 16.0

💖 NAME 💖

okay so this is the /home/abui.../sdrgev3.f(3) page from the Library Functions Manual — and the full NAME path is literally:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/TESTING/EIG/sdrgev3.f

🌸 SYNOPSIS 🌸

✨ Functions/Subroutines ✨

subroutine SDRGEV3 (nsizes, nn, ntypes, dotype, iseed, thresh, nounit,
    a, lda, b, s, t, q, ldq, z, qe, ldqe, alphar, alphai, beta, alphr1,
    alphi1, beta1, work, lwork, result, info)
    SDRGEV3

🦋 Function/Subroutine Documentation 🦋

subroutine SDRGEV3 (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( ldqe, * ) qe, integer ldqe,
    real, dimension( * ) alphar, real, dimension( * ) alphai, real,
    dimension( * ) beta, real, dimension( * ) alphr1, real, dimension( * )
    alphi1, real, dimension( * ) beta1, real, dimension( * ) work, integer
    lwork, real, dimension( * ) result, integer info)
SDRGEV3

✨ Purpose ✨

bestie, SDRGEV3 is the vibe-checker for the nonsymmetric generalized eigenvalue problem driver routine SGGEV3. 💅

SGGEV3 computes, for a pair of n-by-n nonsymmetric matrices (A,B), the generalized eigenvalues and — optionally — the left and right eigenvectors.

A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha/beta = w, such that A - w*B is singular. It is usually represented as the pair (alpha,beta), because there is a reasonable interpretation for beta=0, and even for both being zero. 🫶

A right generalized eigenvector corresponding to a generalized eigenvalue w for a pair of matrices (A,B) is a vector r such that (A - wB) * r = 0. A left generalized eigenvector is a vector l such that l**H * (A - wB) = 0, where l**H is the conjugate-transpose of l.

When SDRGEV3 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 tests will be performed and compared with the threshold THRESH.

Results from SGGEV3:

  1. max over all left eigenvalue/-vector pairs (alpha/beta,l) of
| VL**H * (beta A - alpha B) |/( ulp max(|beta A|, |alpha B|) )

where VL**H is the conjugate-transpose of VL.

  1. | |VL(i)| - 1 | / ulp and whether largest component real

VL(i) denotes the i-th column of VL.

  1. max over all left eigenvalue/-vector pairs (alpha/beta,r) of
| (beta A - alpha B) * VR | / ( ulp max(|beta A|, |alpha B|) )
  1. | |VR(i)| - 1 | / ulp and whether largest component real

VR(i) denotes the i-th column of VR.

  1. W(full) = W(partial)

W(full) denotes the eigenvalues computed when both l and r are also computed, and W(partial) denotes the eigenvalues computed when only W, only W and r, or only W and l are computed.

  1. VL(full) = VL(partial)

VL(full) denotes the left eigenvectors computed when both l and r are computed, and VL(partial) denotes the result when only l is computed.

  1. VR(full) = VR(partial)

VR(full) denotes the right eigenvectors computed when both l and r are also computed, and VR(partial) denotes the result when only l is computed. (yes, that’s what it says — keeping it exactly as the page says it 💀)

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

6.

                                     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
  1. ( D, I ) — where D is diag( 0, 1,..., N-1 ) (a diagonal matrix with those diagonal entries).
  2. ( I, D ).
  3. ( big*D, small*I ) — where big is near overflow and small=1/big.
  4. ( small*D, big*I ).
  5. ( big*I, small*D ).
  6. ( small*I, big*D ).
  7. ( big*D, big*I ).
  8. ( small*D, small*I ).
  9. ( 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.
           t   t
(16) Q ( J , J ) Z     where Q and Z are random orthogonal matrices.
  1. 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 ).
  2. Q ( T1, T2 ) Z — diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 ), diag(T2) = ( 0, 1, 0, 1,..., 1, 0 ), s = machine precision.
  3. 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 )
  1. 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.
  2. Q ( big*T1, small*T2 ) Z — diag(T1) = ( 0, 0, 1, ..., N-3, 0 ), diag(T2) = ( 0, 1, ..., 1, 0, 0 ).
  3. Q ( small*T1, big*T2 ) Z — diag(T1) = ( 0, 0, 1, ..., N-3, 0 ), diag(T2) = ( 0, 1, ..., 1, 0, 0 ).
  4. Q ( small*T1, small*T2 ) Z — diag(T1) = ( 0, 0, 1, ..., N-3, 0 ), diag(T2) = ( 0, 1, ..., 1, 0, 0 ).
  5. Q ( big*T1, big*T2 ) Z — diag(T1) = ( 0, 0, 1, ..., N-3, 0 ), diag(T2) = ( 0, 1, ..., 1, 0, 0 ).
  6. 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, SDRGEV3 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, SDRGEV3 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. 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 SDRGEV3 to continue the same random number sequence.
  • THRESH — THRESH is REAL. A test will count as a fail if the computed value, 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. It must be at least zero.
  • NOUNIT — NOUNIT is INTEGER. The FORTRAN unit number for printing out error messages (e.g., if a routine returns IERR not equal to 0.)
  • A —

okay so continuing the argument list, bestie — keep your eyes on the code, it's all exact, no cap:

  • A — REAL array, dimension (LDA, max(NN)). It holds the original A matrix. It's used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. — that's the whole vibe check, fr fr.
  • LDA — INTEGER. It's the leading dimension of A, B, S, and T. It must be at least 1 and at least max( NN ). Do not mess with this, bestie.
  • B — REAL array, dimension (LDA, max(NN)). It holds the original B matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. Same energy as A, periodt.
  • S — REAL array, dimension (LDA, max(NN)). The Schur form matrix computed from A by SGGEV3. On exit, S contains the Schur form matrix corresponding to the matrix in A. Iconic.
  • T — REAL array, dimension (LDA, max(NN)). The upper triangular matrix computed from B by SGGEV3. Slay.
  • Q — REAL array, dimension (LDQ, max(NN)). The (left) eigenvectors matrix computed by SGGEV3. Main character energy, honestly.
  • LDQ — INTEGER. The leading dimension of Q and Z. It must be at least 1 and at least max( NN ). Highkey important.
  • Z — REAL array, dimension (LDQ, max(NN)). The (right) orthogonal matrix computed by SGGEV3. Obsessed.
  • QE — REAL array, dimension (LDQ, max(NN)). QE holds the computed right or left eigenvectors. Not me forgetting this one.
  • LDQE — INTEGER. The leading dimension of QE. LDQE >= max(1,max(NN)). Literally.
  • ALPHAR — REAL array, dimension (max(NN)). That's it, bestie — just the dimension.
  • ALPHAI — REAL array, dimension (max(NN)). Same vibe, different name.
  • BETA — REAL array, dimension (max(NN)). These are the generalized eigenvalues of (A,B) computed by SGGEV3. The formula is ( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the k-th generalized eigenvalue of A and B. Keep that exact, fr fr.
  • ALPHR1 — REAL array, dimension (max(NN)). Just the dimension, no extra drama.
  • ALPHI1 — REAL array, dimension (max(NN)). Same.
  • BETA1 — REAL array, dimension (max(NN)). Like ALPHAR, ALPHAI, BETA, these arrays contain the eigenvalues of A and B, but those computed when SGGEV3 only computes a partial eigendecomposition, i.e. not the eigenvalues and left and right eigenvectors. Lowkey crucial.
  • WORK — REAL array, dimension (LWORK). The workspace, bestie.
  • LWORK — INTEGER. The number of entries in WORK. LWORK >= MAX( 8*N, N*(N+1) ). Do not skimp.
  • RESULT — REAL array, dimension (2). The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. No cap.
  • 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 404 of file sdrgev3.f. 💅

✨ AUTHOR

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okay bestie, that's the whole page. Footer vibe: LAPACK Version 3.12.0, /home/abui.../sdrgev3.f(3). Stay iconic, and remember: the facts did not change. 💖✨

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