lahqr(3)

NAME
lahqr - lahqr: eig of Hessenberg, step in hseqr

SYNOPSIS

  Functions

subroutine clahqr (wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z, ldz, info)
CLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.
subroutine dlahqr (wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, iloz, ihiz, z, ldz, info)
DLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.
subroutine slahqr (wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, iloz, ihiz, z, ldz, info)
SLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.
subroutine zlahqr (wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z, ldz, info)
ZLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.

Detailed Description

Function Documentation

  subroutine clahqr (logical wantt, logical wantz, integer n, integer ilo, integer ihi, complex, dimension( ldh, * ) h, integer ldh, complex, dimension( * ) w, integer iloz, integer ihiz, complex, dimension( ldz, * ) z, integer ldz, integer info)

CLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.

Purpose:

    !>
    !>    CLAHQR is an auxiliary routine called by CHSEQR to update the
    !>    eigenvalues and Schur decomposition already computed by CHSEQR, by
    !>    dealing with the Hessenberg submatrix in rows and columns ILO to
    !>    IHI.
    !>

Parameters
    WANTT

    !>          WANTT is LOGICAL
    !>          = .TRUE. : the full Schur form T is required;
    !>          = .FALSE.: only eigenvalues are required.
    !>

    WANTZ

    !>          WANTZ is LOGICAL
    !>          = .TRUE. : the matrix of Schur vectors Z is required;
    !>          = .FALSE.: Schur vectors are not required.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix H.  N >= 0.
    !>

    ILO

    !>          ILO is INTEGER
    !>

    IHI

    !>          IHI is INTEGER
    !>          It is assumed that H is already upper triangular in rows and
    !>          columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1).
    !>          CLAHQR works primarily with the Hessenberg submatrix in rows
    !>          and columns ILO to IHI, but applies transformations to all of
    !>          H if WANTT is .TRUE..
    !>          1 <= ILO <= max(1,IHI); IHI <= N.
    !>

    H

    !>          H is COMPLEX array, dimension (LDH,N)
    !>          On entry, the upper Hessenberg matrix H.
    !>          On exit, if INFO is zero and if WANTT is .TRUE., then H
    !>          is upper triangular in rows and columns ILO:IHI.  If INFO
    !>          is zero and if WANTT is .FALSE., then the contents of H
    !>          are unspecified on exit.  The output state of H in case
    !>          INF is positive is below under the description of INFO.
    !>

    LDH

    !>          LDH is INTEGER
    !>          The leading dimension of the array H. LDH >= max(1,N).
    !>

    W

    !>          W is COMPLEX array, dimension (N)
    !>          The computed eigenvalues ILO to IHI are stored in the
    !>          corresponding elements of W. If WANTT is .TRUE., the
    !>          eigenvalues are stored in the same order as on the diagonal
    !>          of the Schur form returned in H, with W(i) = H(i,i).
    !>

    ILOZ

    !>          ILOZ is INTEGER
    !>

    IHIZ

    !>          IHIZ is INTEGER
    !>          Specify the rows of Z to which transformations must be
    !>          applied if WANTZ is .TRUE..
    !>          1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
    !>

    Z

    !>          Z is COMPLEX array, dimension (LDZ,N)
    !>          If WANTZ is .TRUE., on entry Z must contain the current
    !>          matrix Z of transformations accumulated by CHSEQR, and on
    !>          exit Z has been updated; transformations are applied only to
    !>          the submatrix Z(ILOZ:IHIZ,ILO:IHI).
    !>          If WANTZ is .FALSE., Z is not referenced.
    !>

    LDZ

    !>          LDZ is INTEGER
    !>          The leading dimension of the array Z. LDZ >= max(1,N).
    !>

    INFO

    !>          INFO is INTEGER
    !>           = 0:  successful exit
    !>           > 0:  if INFO = i, CLAHQR failed to compute all the
    !>                  eigenvalues ILO to IHI in a total of 30 iterations
    !>                  per eigenvalue; elements i+1:ihi of W contain
    !>                  those eigenvalues which have been successfully
    !>                  computed.
    !>
    !>                  If INFO > 0 and WANTT is .FALSE., then on exit,
    !>                  the remaining unconverged eigenvalues are the
    !>                  eigenvalues of the upper Hessenberg matrix
    !>                  rows and columns ILO through INFO of the final,
    !>                  output value of H.
    !>
    !>                  If INFO > 0 and WANTT is .TRUE., then on exit
    !>          (*)       (initial value of H)*U  = U*(final value of H)
    !>                  where U is an orthogonal matrix.    The final
    !>                  value of H is upper Hessenberg and triangular in
    !>                  rows and columns INFO+1 through IHI.
    !>
    !>                  If INFO > 0 and WANTZ is .TRUE., then on exit
    !>                      (final value of Z)  = (initial value of Z)*U
    !>                  where U is the orthogonal matrix in (*)
    !>                  (regardless of the value of WANTT.)
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

    !>
    !>     02-96 Based on modifications by
    !>     David Day, Sandia National Laboratory, USA
    !>
    !>     12-04 Further modifications by
    !>     Ralph Byers, University of Kansas, USA
    !>     This is a modified version of CLAHQR from LAPACK version 3.0.
    !>     It is (1) more robust against overflow and underflow and
    !>     (2) adopts the more conservative Ahues & Tisseur stopping
    !>     criterion (LAWN 122, 1997).
    !>

  subroutine dlahqr (logical wantt, logical wantz, integer n, integer ilo, integer ihi, double precision, dimension( ldh, * ) h, integer ldh, double precision, dimension( * ) wr, double precision, dimension( * ) wi, integer iloz, integer ihiz, double precision, dimension( ldz, * ) z, integer ldz, integer info)

DLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.

Purpose:

    !>
    !>    DLAHQR is an auxiliary routine called by DHSEQR to update the
    !>    eigenvalues and Schur decomposition already computed by DHSEQR, by
    !>    dealing with the Hessenberg submatrix in rows and columns ILO to
    !>    IHI.
    !>

Parameters
    WANTT

    !>          WANTT is LOGICAL
    !>          = .TRUE. : the full Schur form T is required;
    !>          = .FALSE.: only eigenvalues are required.
    !>

    WANTZ

    !>          WANTZ is LOGICAL
    !>          = .TRUE. : the matrix of Schur vectors Z is required;
    !>          = .FALSE.: Schur vectors are not required.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix H.  N >= 0.
    !>

    ILO

    !>          ILO is INTEGER
    !>

    IHI

    !>          IHI is INTEGER
    !>          It is assumed that H is already upper quasi-triangular in
    !>          rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless
    !>          ILO = 1). DLAHQR works primarily with the Hessenberg
    !>          submatrix in rows and columns ILO to IHI, but applies
    !>          transformations to all of H if WANTT is .TRUE..
    !>          1 <= ILO <= max(1,IHI); IHI <= N.
    !>

    H

    !>          H is DOUBLE PRECISION array, dimension (LDH,N)
    !>          On entry, the upper Hessenberg matrix H.
    !>          On exit, if INFO is zero and if WANTT is .TRUE., H is upper
    !>          quasi-triangular in rows and columns ILO:IHI, with any
    !>          2-by-2 diagonal blocks in standard form. If INFO is zero
    !>          and WANTT is .FALSE., the contents of H are unspecified on
    !>          exit.  The output state of H if INFO is nonzero is given
    !>          below under the description of INFO.
    !>

    LDH

    !>          LDH is INTEGER
    !>          The leading dimension of the array H. LDH >= max(1,N).
    !>

    WR

    !>          WR is DOUBLE PRECISION array, dimension (N)
    !>

    WI

    !>          WI is DOUBLE PRECISION array, dimension (N)
    !>          The real and imaginary parts, respectively, of the computed
    !>          eigenvalues ILO to IHI are stored in the corresponding
    !>          elements of WR and WI. If two eigenvalues are computed as a
    !>          complex conjugate pair, they are stored in consecutive
    !>          elements of WR and WI, say the i-th and (i+1)th, with
    !>          WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the
    !>          eigenvalues are stored in the same order as on the diagonal
    !>          of the Schur form returned in H, with WR(i) = H(i,i), and, if
    !>          H(i:i+1,i:i+1) is a 2-by-2 diagonal block,
    !>          WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).
    !>

    ILOZ

    !>          ILOZ is INTEGER
    !>

    IHIZ

    !>          IHIZ is INTEGER
    !>          Specify the rows of Z to which transformations must be
    !>          applied if WANTZ is .TRUE..
    !>          1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
    !>

    Z

    !>          Z is DOUBLE PRECISION array, dimension (LDZ,N)
    !>          If WANTZ is .TRUE., on entry Z must contain the current
    !>          matrix Z of transformations accumulated by DHSEQR, and on
    !>          exit Z has been updated; transformations are applied only to
    !>          the submatrix Z(ILOZ:IHIZ,ILO:IHI).
    !>          If WANTZ is .FALSE., Z is not referenced.
    !>

    LDZ

    !>          LDZ is INTEGER
    !>          The leading dimension of the array Z. LDZ >= max(1,N).
    !>

    INFO

    !>          INFO is INTEGER
    !>           = 0:  successful exit
    !>           > 0:  If INFO = i, DLAHQR failed to compute all the
    !>                  eigenvalues ILO to IHI in a total of 30 iterations
    !>                  per eigenvalue; elements i+1:ihi of WR and WI
    !>                  contain those eigenvalues which have been
    !>                  successfully computed.
    !>
    !>                  If INFO > 0 and WANTT is .FALSE., then on exit,
    !>                  the remaining unconverged eigenvalues are the
    !>                  eigenvalues of the upper Hessenberg matrix rows
    !>                  and columns ILO through INFO of the final, output
    !>                  value of H.
    !>
    !>                  If INFO > 0 and WANTT is .TRUE., then on exit
    !>          (*)       (initial value of H)*U  = U*(final value of H)
    !>                  where U is an orthogonal matrix.    The final
    !>                  value of H is upper Hessenberg and triangular in
    !>                  rows and columns INFO+1 through IHI.
    !>
    !>                  If INFO > 0 and WANTZ is .TRUE., then on exit
    !>                      (final value of Z)  = (initial value of Z)*U
    !>                  where U is the orthogonal matrix in (*)
    !>                  (regardless of the value of WANTT.)
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>     02-96 Based on modifications by
    !>     David Day, Sandia National Laboratory, USA
    !>
    !>     12-04 Further modifications by
    !>     Ralph Byers, University of Kansas, USA
    !>     This is a modified version of DLAHQR from LAPACK version 3.0.
    !>     It is (1) more robust against overflow and underflow and
    !>     (2) adopts the more conservative Ahues & Tisseur stopping
    !>     criterion (LAWN 122, 1997).
    !>

  subroutine slahqr (logical wantt, logical wantz, integer n, integer ilo, integer ihi, real, dimension( ldh, * ) h, integer ldh, real, dimension( * ) wr, real, dimension( * ) wi, integer iloz, integer ihiz, real, dimension( ldz, * ) z, integer ldz, integer info)

SLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.

Purpose:

    !>
    !>    SLAHQR is an auxiliary routine called by SHSEQR to update the
    !>    eigenvalues and Schur decomposition already computed by SHSEQR, by
    !>    dealing with the Hessenberg submatrix in rows and columns ILO to
    !>    IHI.
    !>

Parameters
    WANTT

    !>          WANTT is LOGICAL
    !>          = .TRUE. : the full Schur form T is required;
    !>          = .FALSE.: only eigenvalues are required.
    !>

    WANTZ

    !>          WANTZ is LOGICAL
    !>          = .TRUE. : the matrix of Schur vectors Z is required;
    !>          = .FALSE.: Schur vectors are not required.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix H.  N >= 0.
    !>

    ILO

    !>          ILO is INTEGER
    !>

    IHI

    !>          IHI is INTEGER
    !>          It is assumed that H is already upper quasi-triangular in
    !>          rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless
    !>          ILO = 1). SLAHQR works primarily with the Hessenberg
    !>          submatrix in rows and columns ILO to IHI, but applies
    !>          transformations to all of H if WANTT is .TRUE..
    !>          1 <= ILO <= max(1,IHI); IHI <= N.
    !>

    H

    !>          H is REAL array, dimension (LDH,N)
    !>          On entry, the upper Hessenberg matrix H.
    !>          On exit, if INFO is zero and if WANTT is .TRUE., H is upper
    !>          quasi-triangular in rows and columns ILO:IHI, with any
    !>          2-by-2 diagonal blocks in standard form. If INFO is zero
    !>          and WANTT is .FALSE., the contents of H are unspecified on
    !>          exit.  The output state of H if INFO is nonzero is given
    !>          below under the description of INFO.
    !>

    LDH

    !>          LDH is INTEGER
    !>          The leading dimension of the array H. LDH >= max(1,N).
    !>

    WR

    !>          WR is REAL array, dimension (N)
    !>

    WI

    !>          WI is REAL array, dimension (N)
    !>          The real and imaginary parts, respectively, of the computed
    !>          eigenvalues ILO to IHI are stored in the corresponding
    !>          elements of WR and WI. If two eigenvalues are computed as a
    !>          complex conjugate pair, they are stored in consecutive
    !>          elements of WR and WI, say the i-th and (i+1)th, with
    !>          WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the
    !>          eigenvalues are stored in the same order as on the diagonal
    !>          of the Schur form returned in H, with WR(i) = H(i,i), and, if
    !>          H(i:i+1,i:i+1) is a 2-by-2 diagonal block,
    !>          WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).
    !>

    ILOZ

    !>          ILOZ is INTEGER
    !>

    IHIZ

    !>          IHIZ is INTEGER
    !>          Specify the rows of Z to which transformations must be
    !>          applied if WANTZ is .TRUE..
    !>          1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
    !>

    Z

    !>          Z is REAL array, dimension (LDZ,N)
    !>          If WANTZ is .TRUE., on entry Z must contain the current
    !>          matrix Z of transformations accumulated by SHSEQR, and on
    !>          exit Z has been updated; transformations are applied only to
    !>          the submatrix Z(ILOZ:IHIZ,ILO:IHI).
    !>          If WANTZ is .FALSE., Z is not referenced.
    !>

    LDZ

    !>          LDZ is INTEGER
    !>          The leading dimension of the array Z. LDZ >= max(1,N).
    !>

    INFO

    !>          INFO is INTEGER
    !>           = 0:   successful exit
    !>           > 0:   If INFO = i, SLAHQR failed to compute all the
    !>                  eigenvalues ILO to IHI in a total of 30 iterations
    !>                  per eigenvalue; elements i+1:ihi of WR and WI
    !>                  contain those eigenvalues which have been
    !>                  successfully computed.
    !>
    !>                  If INFO > 0 and WANTT is .FALSE., then on exit,
    !>                  the remaining unconverged eigenvalues are the
    !>                  eigenvalues of the upper Hessenberg matrix rows
    !>                  and columns ILO through INFO of the final, output
    !>                  value of H.
    !>
    !>                  If INFO > 0 and WANTT is .TRUE., then on exit
    !>          (*)       (initial value of H)*U  = U*(final value of H)
    !>                  where U is an orthogonal matrix.    The final
    !>                  value of H is upper Hessenberg and triangular in
    !>                  rows and columns INFO+1 through IHI.
    !>
    !>                  If INFO > 0 and WANTZ is .TRUE., then on exit
    !>                      (final value of Z)  = (initial value of Z)*U
    !>                  where U is the orthogonal matrix in (*)
    !>                  (regardless of the value of WANTT.)
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>     02-96 Based on modifications by
    !>     David Day, Sandia National Laboratory, USA
    !>
    !>     12-04 Further modifications by
    !>     Ralph Byers, University of Kansas, USA
    !>     This is a modified version of SLAHQR from LAPACK version 3.0.
    !>     It is (1) more robust against overflow and underflow and
    !>     (2) adopts the more conservative Ahues & Tisseur stopping
    !>     criterion (LAWN 122, 1997).
    !>

  subroutine zlahqr (logical wantt, logical wantz, integer n, integer ilo, integer ihi, complex*16, dimension( ldh, * ) h, integer ldh, complex*16, dimension( * ) w, integer iloz, integer ihiz, complex*16, dimension( ldz, * ) z, integer ldz, integer info)

ZLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.

Purpose:

    !>
    !>    ZLAHQR is an auxiliary routine called by CHSEQR to update the
    !>    eigenvalues and Schur decomposition already computed by CHSEQR, by
    !>    dealing with the Hessenberg submatrix in rows and columns ILO to
    !>    IHI.
    !>

Parameters
    WANTT

    !>          WANTT is LOGICAL
    !>          = .TRUE. : the full Schur form T is required;
    !>          = .FALSE.: only eigenvalues are required.
    !>

    WANTZ

    !>          WANTZ is LOGICAL
    !>          = .TRUE. : the matrix of Schur vectors Z is required;
    !>          = .FALSE.: Schur vectors are not required.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix H.  N >= 0.
    !>

    ILO

    !>          ILO is INTEGER
    !>

    IHI

    !>          IHI is INTEGER
    !>          It is assumed that H is already upper triangular in rows and
    !>          columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1).
    !>          ZLAHQR works primarily with the Hessenberg submatrix in rows
    !>          and columns ILO to IHI, but applies transformations to all of
    !>          H if WANTT is .TRUE..
    !>          1 <= ILO <= max(1,IHI); IHI <= N.
    !>

    H

    !>          H is COMPLEX*16 array, dimension (LDH,N)
    !>          On entry, the upper Hessenberg matrix H.
    !>          On exit, if INFO is zero and if WANTT is .TRUE., then H
    !>          is upper triangular in rows and columns ILO:IHI.  If INFO
    !>          is zero and if WANTT is .FALSE., then the contents of H
    !>          are unspecified on exit.  The output state of H in case
    !>          INF is positive is below under the description of INFO.
    !>

    LDH

    !>          LDH is INTEGER
    !>          The leading dimension of the array H. LDH >= max(1,N).
    !>

    W

    !>          W is COMPLEX*16 array, dimension (N)
    !>          The computed eigenvalues ILO to IHI are stored in the
    !>          corresponding elements of W. If WANTT is .TRUE., the
    !>          eigenvalues are stored in the same order as on the diagonal
    !>          of the Schur form returned in H, with W(i) = H(i,i).
    !>

    ILOZ

    !>          ILOZ is INTEGER
    !>

    IHIZ

    !>          IHIZ is INTEGER
    !>          Specify the rows of Z to which transformations must be
    !>          applied if WANTZ is .TRUE..
    !>          1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
    !>

    Z

    !>          Z is COMPLEX*16 array, dimension (LDZ,N)
    !>          If WANTZ is .TRUE., on entry Z must contain the current
    !>          matrix Z of transformations accumulated by CHSEQR, and on
    !>          exit Z has been updated; transformations are applied only to
    !>          the submatrix Z(ILOZ:IHIZ,ILO:IHI).
    !>          If WANTZ is .FALSE., Z is not referenced.
    !>

    LDZ

    !>          LDZ is INTEGER
    !>          The leading dimension of the array Z. LDZ >= max(1,N).
    !>

    INFO

    !>          INFO is INTEGER
    !>           = 0:   successful exit
    !>           > 0:   if INFO = i, ZLAHQR failed to compute all the
    !>                  eigenvalues ILO to IHI in a total of 30 iterations
    !>                  per eigenvalue; elements i+1:ihi of W contain
    !>                  those eigenvalues which have been successfully
    !>                  computed.
    !>
    !>                  If INFO > 0 and WANTT is .FALSE., then on exit,
    !>                  the remaining unconverged eigenvalues are the
    !>                  eigenvalues of the upper Hessenberg matrix
    !>                  rows and columns ILO through INFO of the final,
    !>                  output value of H.
    !>
    !>                  If INFO > 0 and WANTT is .TRUE., then on exit
    !>          (*)       (initial value of H)*U  = U*(final value of H)
    !>                  where U is an orthogonal matrix.    The final
    !>                  value of H is upper Hessenberg and triangular in
    !>                  rows and columns INFO+1 through IHI.
    !>
    !>                  If INFO > 0 and WANTZ is .TRUE., then on exit
    !>                      (final value of Z)  = (initial value of Z)*U
    !>                  where U is the orthogonal matrix in (*)
    !>                  (regardless of the value of WANTT.)
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

    !>
    !>     02-96 Based on modifications by
    !>     David Day, Sandia National Laboratory, USA
    !>
    !>     12-04 Further modifications by
    !>     Ralph Byers, University of Kansas, USA
    !>     This is a modified version of ZLAHQR from LAPACK version 3.0.
    !>     It is (1) more robust against overflow and underflow and
    !>     (2) adopts the more conservative Ahues & Tisseur stopping
    !>     criterion (LAWN 122, 1997).
    !>

Author

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