TESTING/EIG/sdrgev3.f(3)

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🎀 TESTING/EIG/sdrgev3.f(3) — Library Functions Manual 🎀

bestie, this is the sdrgev3 page and it is long, so we’re doing it in parts. Part 1 starts now. No cap, every flag, formula, and matrix type is staying. 💅

💖 NAME 💖

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

Yep, the main character is SDRGEV3 itself. 👑

🌸 Purpose 🌸

Okay so SDRGEV3 is the vibe check 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), as there is 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 () 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 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.

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

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

(3)  max over all left eigenvalue/-vector pairs (alpha/beta,r) of

     | (beta A - alpha B) * VR | / ( ulp max(|beta A|, |alpha B|) )

(4)  | |VR(i)| - 1 | / ulp and whether largest component real

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

(5)  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.

(6)  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.

(7)  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.

🧪 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)

        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  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. It’s 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 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. 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

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 — INTEGER. It’s the leading dimension of A, B, S, and T. It must be at least 1 and at least max( NN ). That’s the rule, bestie 💅
  • B — REAL array, dimension(LDA, max(NN)). Used to hold the original B matrix. It’s used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. — very specific vibe check, no cap 🎀
  • 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 ✨
  • T — REAL array, dimension (LDA, max(NN)). The upper triangular matrix computed from B by SGGEV3 🦋
  • Q — REAL array, dimension (LDQ, max(NN)). The (left) eigenvectors matrix computed by SGGEV3 💖
  • LDQ — INTEGER. The leading dimension of Q and Z. It must be at least 1 and at least max( NN ) 🌸
  • Z — REAL array, dimension( LDQ, max(NN) ). The (right) orthogonal matrix computed by SGGEV3 👑
  • QE — REAL array, dimension( LDQ, max(NN) ). QE holds the computed right or left eigenvectors 🫶
  • LDQE — INTEGER. The leading dimension of QE. LDQE >= max(1,max(NN)) ⭐
  • ALPHAR — REAL array, dimension (max(NN)) 💜
  • ALPHAI — REAL array, dimension (max(NN)) 💜
  • BETA — REAL array, dimension (max(NN)). The generalized eigenvalues of (A,B) computed by SGGEV3. ( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the k-th generalized eigenvalue of A and B 🔥
  • ALPHR1 — REAL array, dimension (max(NN)) 🍓
  • ALPHI1 — REAL array, dimension (max(NN)) 🍓
  • 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. That distinction is lowkey important, bestie 😭
  • WORK — REAL array, dimension (LWORK) 🌈
  • LWORK — INTEGER. The number of entries in WORK. LWORK >= MAX( 8*N, N*(N+1) ). No skimping on the work array, periodt 💀
  • RESULT — REAL array, dimension (2). The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow 🙈
  • INFO — INTEGER. Here’s the exit-status vibe check:
    • = 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 👑

These absolute icons co-signed:

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

Definition at line 404 of file sdrgev3.f.

🤖 Author (auto-generated) 🤖

Generated automatically by Doxygen for LAPACK from the source code.

💖 That’s the whole parameter glow-up, bestie. Source: LAPACK. Version: 3.12.0. Date: not listed on this page, so I’m not inventing one, periodt. File: TESTING/EIG/sdrgev3.f(3). 🎀

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