TESTING/EIG/dget23.f(3)

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💅 TESTING/EIG/dget23.f — Library Functions Manual 💅

okay bestie, welcome to the Library Functions Manual, specifically the page for TESTING/EIG/dget23.f. This is the DGET23 saga, and it is giving “Fortran eigenvalue-testing main character” energy. No cap. 💖

🌸 NAME 🌸

TESTING/EIG/dget23.f — that’s the file, that’s the name, periodt. 🎀

✨ SYNOPSIS ✨

🎀 Functions/Subroutines 🎀

  • subroutine dget23 — here’s the whole call, bestie, keep it exact:
subroutine dget23 (comp, balanc, jtype, thresh, iseed, nounit, n, a,
    lda, h, wr, wi, wr1, wi1, vl, ldvl, vr, ldvr, lre, ldlre, rcondv,
    rcndv1, rcdvin, rconde, rcnde1, rcdein, scale, scale1, result, work,
    lwork, iwork, info)
  • DGET23 — the name, the icon, the moment. 👑

🌈 Function/Subroutine Documentation 🌈

💖 subroutine dget23 💖

subroutine dget23 (logical comp, character balanc, integer jtype, double
    precision thresh, integer, dimension( 4 ) iseed, integer nounit, integer
    n, double precision, dimension( lda, * ) a, integer lda, double
    precision, dimension( lda, * ) h, double precision, dimension( * ) wr,
    double precision, dimension( * ) wi, double precision, dimension( * )
    wr1, double precision, dimension( * ) wi1, double precision, dimension(
    ldvl, * ) vl, integer ldvl, double precision, dimension( ldvr, * ) vr,
    integer ldvr, double precision, dimension( ldlre, * ) lre, integer
    ldlre, double precision, dimension( * ) rcondv, double precision,
    dimension( * ) rcndv1, double precision, dimension( * ) rcdvin, double
    precision, dimension( * ) rconde, double precision, dimension( * )
    rcnde1, double precision, dimension( * ) rcdein, double precision,
    dimension( * ) scale, double precision, dimension( * ) scale1, double
    precision, dimension( 11 ) result, double precision, dimension( * )
    work, integer lwork, integer, dimension( * ) iwork, integer info)
    DGET23

DGET23 — yes, she appears again. Main character energy. 💅

🦋 Purpose 🦋

DGET23 checks the nonsymmetric eigenvalue problem driver SGEEVX.

If COMP = .FALSE., the first 8 of the following tests will be performed on the input matrix A, and also test 9 if LWORK is sufficiently large. If COMP is .TRUE., all 11 tests will be performed. 🫶

  • (1) — | A * VR - VR * W | / ( n |A| ulp )

Here VR is the matrix of unit right eigenvectors. W is a block diagonal matrix, with a 1x1 block for each real eigenvalue and a 2x2 block for each complex conjugate pair. If eigenvalues j and j+1 are a complex conjugate pair, so WR(j) = WR(j+1) = wr and WI(j) = - WI(j+1) = wi, then the 2 x 2 block corresponding to the pair will be:

(  wr  wi  )
( -wi  wr  )

Such a block multiplying an n x 2 matrix ( ur ui ) on the right will be the same as multiplying ur + i*ui by wr + i*wi.

  • (2) — | A**H * VL - VL * W**H | / ( n |A| ulp )

Here VL is the matrix of unit left eigenvectors, A**H is the conjugate transpose of A, and W is as above.

  • (3) — | |VR(i)| - 1 | / ulp and largest component real

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

  • (4) — | |VL(i)| - 1 | / ulp and largest component real

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

  • (5) — 0 if W(full) = W(partial), 1/ulp otherwise

W(full) denotes the eigenvalues computed when VR, VL, RCONDV and RCONDE are also computed, and W(partial) denotes the eigenvalues computed when only some of VR, VL, RCONDV, and RCONDE are computed.

  • (6) — 0 if VR(full) = VR(partial), 1/ulp otherwise

VR(full) denotes the right eigenvectors computed when VL, RCONDV and RCONDE are computed, and VR(partial) denotes the result when only some of VL and RCONDV are computed.

  • (7) — 0 if VL(full) = VL(partial), 1/ulp otherwise

VL(full) denotes the left eigenvectors computed when VR, RCONDV and RCONDE are computed, and VL(partial) denotes the result when only some of VR and RCONDV are computed.

  • (8) — 0 if SCALE, ILO, IHI, ABNRM (full) = SCALE, ILO, IHI, ABNRM (partial) 1/ulp otherwise

SCALE, ILO, IHI and ABNRM describe how the matrix is balanced. (full) is when VR, VL, RCONDE and RCONDV are also computed, and (partial) is when some are not computed.

  • (9) — 0 if RCONDV(full) = RCONDV(partial), 1/ulp otherwise

RCONDV(full) denotes the reciprocal condition numbers of the right eigenvectors computed when VR, VL and RCONDE are also computed. RCONDV(partial) denotes the reciprocal condition numbers when only some of VR, VL and RCONDE are computed.

  • (10) — |RCONDV - RCDVIN| / cond(RCONDV)

RCONDV is the reciprocal right eigenvector condition number computed by DGEEVX and RCDVIN (the precomputed true value) is supplied as input. cond(RCONDV) is the condition number of RCONDV, and takes errors in computing RCONDV into account, so that the resulting quantity should be O(ULP). cond(RCONDV) is essentially given by norm(A)/RCONDE.

  • (11) — |RCONDE - RCDEIN| / cond(RCONDE)

RCONDE is the reciprocal eigenvalue condition number computed by DGEEVX and RCDEIN (the precomputed true value) is supplied as input. cond(RCONDE) is the condition number of RCONDE, and takes errors in computing RCONDE into account, so that the resulting quantity should be O(ULP). cond(RCONDE) is essentially given by norm(A)/RCONDV.

🫶 Parameters 🫶

okay bestie, parameter time. Every single one is here, no skipping. 💜

COMP

COMP is LOGICAL.

COMP describes which input tests to perform:

  • = .FALSE. if the computed condition numbers are not to be tested against RCDVIN and RCDEIN
  • = .TRUE. if they are to be compared

BALANC

BALANC is CHARACTER.

Describes the balancing option to be tested.

  • = 'N' for no permuting or diagonal scaling
  • = 'P' for permuting but no diagonal scaling
  • = 'S' for no permuting but diagonal scaling
  • = 'B' for permuting and diagonal scaling

JTYPE

JTYPE is INTEGER.

Type of input matrix. Used to label output if error occurs.

THRESH

THRESH is DOUBLE PRECISION.

A test will count as a fail 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. It must be at least zero.

bestie, “at least zero” is not a suggestion. It is the rule. ⚠️

ISEED

ISEED is INTEGER array, dimension (4).

If COMP = .FALSE., the random number generator seed used to produce matrix. If COMP = .TRUE., ISEED(1) = the number of the example. Used to label output if error occurs.

NOUNIT

NOUNIT is INTEGER.

The FORTRAN unit number for printing out error messages (e.g., if a routine returns INFO not equal to 0.)

N

N is INTEGER.

The dimension of A. N must be at least 0.

A

A is DOUBLE PRECISION array, dimension (LDA,N).

Used to hold the matrix whose eigenvalues are to be computed.

LDA

LDA is INTEGER.

The leading dimension of A, and H. LDA must be at least 1 and at least N.

H

H is DOUBLE PRECISION array, dimension (LDA,N).

Another copy of the test matrix A, modified by DGEEVX.

WR

WR is DOUBLE PRECISION array, dimension (N).

WI

WI is DOUBLE PRECISION array, dimension (N).

The real and imaginary parts of the eigenvalues of A. On exit, WR + WI*i are the eigenvalues of the matrix in A.

WR1

WR1 is DOUBLE PRECISION array, dimension (N).

WI1

WI1 is DOUBLE PRECISION array, dimension (N).

Like WR, WI, these arrays contain the eigenvalues of A, but those computed when DGEEVX only computes a partial eigendecomposition, i.e. not the eigenvalues and left and right eigenvectors.

VL

VL is DOUBLE PRECISION array, dimension (LDVL,N).

VL holds the computed left eigenvectors.

LDVL

LDVL is INTEGER.

Leading dimension of VL. Must be at least max(1,N).

VR

VR is DOUBLE PRECISION array, dimension (LDVR,N).

VR holds the computed right eigenvectors.

LDVR

LDVR is INTEGER.

Leading dimension of VR. Must be at least max(1,N).

LRE

LRE is DOUBLE PRECISION array, dimension (LDLRE,N).

LRE holds the computed right or left eigenvectors.

LDLRE

LDLRE is INTEGER.

Leading dimension of LRE. Must be at least max(1,N).

RCONDV

RCONDV is DOUBLE PRECISION array, dimension (N).

RCONDV holds the computed reciprocal condition numbers for eigenvectors.

RCNDV1

RCNDV1 is DOUBLE PRECISION array, dimension (N).

RCNDV1 holds more computed reciprocal condition numbers for eigenvectors.

RCDVIN

RCDVIN is DOUBLE PRECISION array, dimension (N).

When COMP = .TRUE. RCDVIN holds the precomputed reciprocal condition numbers for eigenvectors to be compared with RCONDV.

RCONDE

RCONDE is DOUBLE PRECISION array, dimension (N).

RCONDE holds the computed reciprocal condition numbers for eigenvalues.

RCNDE1

The rest of the parameter crew is right here — every type, dimension, and edge case is staying, no cap 👑:

  • RCNDE1 — it's a DOUBLE PRECISION array, dimension (N). It holds more computed reciprocal condition numbers for eigenvalues. 💖
  • RCDEIN — also a DOUBLE PRECISION array, dimension (N). When COMP = .TRUE., RCDEIN holds the precomputed reciprocal condition numbers for eigenvalues to be compared with RCONDE. ✨
  • SCALE — a DOUBLE PRECISION array, dimension (N). Holds information describing balancing of matrix. 🎀
  • SCALE1 — a DOUBLE PRECISION array, dimension (N). Holds information describing balancing of matrix. 🦋
  • RESULT — a DOUBLE PRECISION array, dimension (11). These are the values computed by the 11 tests described above. The values are currently limited to 1/ulp, to avoid overflow. 🌸
  • WORK — a DOUBLE PRECISION array, dimension (LWORK). 💅
  • LWORK — an INTEGER. The number of entries in WORK. This must be at least 3*N, and 6*N+N**2 if tests 9, 10 or 11 are to be performed. 🫶
  • IWORK — an INTEGER array, dimension (2*N). 😭
  • INFO — an INTEGER. Here's the vibe check:
    • If 0, successful exit.
    • If <0, input parameter -INFO had an incorrect value.
    • If >0, DGEEVX returned an error code, the absolute value of which is returned. 🔥

👑 Author

The author squad is:

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

And the source says: Definition at line 373 of file dget23.f. 👑

🤖 Author

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