Aggressive early deflation:


    SLAQR3 accepts as input an upper Hessenberg matrix


    H and performs an orthogonal similarity transformation


    designed to detect and deflate fully converged eigenvalues from


    a trailing principal submatrix.  On output H has been over-


    written by a new Hessenberg matrix that is a perturbation of


    an orthogonal similarity transformation of H.  It is to be


    hoped that the final version of H has many zero subdiagonal


    entries.

WANTT

          WANTT is LOGICAL


          If .TRUE., then the Hessenberg matrix H is fully updated


          so that the quasi-triangular Schur factor may be


          computed (in cooperation with the calling subroutine).


          If .FALSE., then only enough of H is updated to preserve


          the eigenvalues.

WANTZ

          WANTZ is LOGICAL


          If .TRUE., then the orthogonal matrix Z is updated so


          so that the orthogonal Schur factor may be computed


          (in cooperation with the calling subroutine).


          If .FALSE., then Z is not referenced.

N

          N is INTEGER


          The order of the matrix H and (if WANTZ is .TRUE.) the


          order of the orthogonal matrix Z.

KTOP

          KTOP is INTEGER


          It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.


          KBOT and KTOP together determine an isolated block


          along the diagonal of the Hessenberg matrix.

KBOT

          KBOT is INTEGER


          It is assumed without a check that either


          KBOT = N or H(KBOT+1,KBOT)=0.  KBOT and KTOP together


          determine an isolated block along the diagonal of the


          Hessenberg matrix.

NW

          NW is INTEGER


          Deflation window size.  1 .LE. NW .LE. (KBOT-KTOP+1).

H

          H is REAL array, dimension (LDH,N)


          On input the initial N-by-N section of H stores the


          Hessenberg matrix undergoing aggressive early deflation.


          On output H has been transformed by an orthogonal


          similarity transformation, perturbed, and the returned


          to Hessenberg form that (it is to be hoped) has some


          zero subdiagonal entries.

LDH

          LDH is INTEGER


          Leading dimension of H just as declared in the calling


          subroutine.  N .LE. LDH

ILOZ

          ILOZ is INTEGER

IHIZ

          IHIZ is INTEGER


          Specify the rows of Z to which transformations must be


          applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.

Z

          Z is REAL array, dimension (LDZ,N)


          IF WANTZ is .TRUE., then on output, the orthogonal


          similarity transformation mentioned above has been


          accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.


          If WANTZ is .FALSE., then Z is unreferenced.

LDZ

          LDZ is INTEGER


          The leading dimension of Z just as declared in the


          calling subroutine.  1 .LE. LDZ.

NS

          NS is INTEGER


          The number of unconverged (ie approximate) eigenvalues


          returned in SR and SI that may be used as shifts by the


          calling subroutine.

ND

          ND is INTEGER


          The number of converged eigenvalues uncovered by this


          subroutine.

SR

          SR is REAL array, dimension (KBOT)

SI

          SI is REAL array, dimension (KBOT)


          On output, the real and imaginary parts of approximate


          eigenvalues that may be used for shifts are stored in


          SR(KBOT-ND-NS+1) through SR(KBOT-ND) and


          SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.


          The real and imaginary parts of converged eigenvalues


          are stored in SR(KBOT-ND+1) through SR(KBOT) and


          SI(KBOT-ND+1) through SI(KBOT), respectively.

V

          V is REAL array, dimension (LDV,NW)


          An NW-by-NW work array.

LDV

          LDV is INTEGER


          The leading dimension of V just as declared in the


          calling subroutine.  NW .LE. LDV

NH

          NH is INTEGER


          The number of columns of T.  NH.GE.NW.

T

          T is REAL array, dimension (LDT,NW)

LDT

          LDT is INTEGER


          The leading dimension of T just as declared in the


          calling subroutine.  NW .LE. LDT

NV

          NV is INTEGER


          The number of rows of work array WV available for


          workspace.  NV.GE.NW.

WV

          WV is REAL array, dimension (LDWV,NW)

LDWV

          LDWV is INTEGER


          The leading dimension of W just as declared in the


          calling subroutine.  NW .LE. LDV

WORK

          WORK is REAL array, dimension (LWORK)


          On exit, WORK(1) is set to an estimate of the optimal value


          of LWORK for the given values of N, NW, KTOP and KBOT.

LWORK

          LWORK is INTEGER


          The dimension of the work array WORK.  LWORK = 2*NW


          suffices, but greater efficiency may result from larger


          values of LWORK.


          If LWORK = -1, then a workspace query is assumed; SLAQR3


          only estimates the optimal workspace size for the given


          values of N, NW, KTOP and KBOT.  The estimate is returned


          in WORK(1).  No error message related to LWORK is issued


          by XERBLA.  Neither H nor Z are accessed.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

June 2016

Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA