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laqr5(3) LAPACK laqr5(3)

NAME

laqr5 - laqr5: step in hseqr

SYNOPSIS

Functions


subroutine claqr5 (wantt, wantz, kacc22, n, ktop, kbot, nshfts, s, h, ldh, iloz, ihiz, z, ldz, v, ldv, u, ldu, nv, wv, ldwv, nh, wh, ldwh)
CLAQR5 performs a single small-bulge multi-shift QR sweep. subroutine dlaqr5 (wantt, wantz, kacc22, n, ktop, kbot, nshfts, sr, si, h, ldh, iloz, ihiz, z, ldz, v, ldv, u, ldu, nv, wv, ldwv, nh, wh, ldwh)
DLAQR5 performs a single small-bulge multi-shift QR sweep. subroutine slaqr5 (wantt, wantz, kacc22, n, ktop, kbot, nshfts, sr, si, h, ldh, iloz, ihiz, z, ldz, v, ldv, u, ldu, nv, wv, ldwv, nh, wh, ldwh)
SLAQR5 performs a single small-bulge multi-shift QR sweep. subroutine zlaqr5 (wantt, wantz, kacc22, n, ktop, kbot, nshfts, s, h, ldh, iloz, ihiz, z, ldz, v, ldv, u, ldu, nv, wv, ldwv, nh, wh, ldwh)
ZLAQR5 performs a single small-bulge multi-shift QR sweep.

Detailed Description

Function Documentation

subroutine claqr5 (logical wantt, logical wantz, integer kacc22, integer n, integer ktop, integer kbot, integer nshfts, complex, dimension( * ) s, complex, dimension( ldh, * ) h, integer ldh, integer iloz, integer ihiz, complex, dimension( ldz, * ) z, integer ldz, complex, dimension( ldv, * ) v, integer ldv, complex, dimension( ldu, * ) u, integer ldu, integer nv, complex, dimension( ldwv, * ) wv, integer ldwv, integer nh, complex, dimension( ldwh, * ) wh, integer ldwh)

CLAQR5 performs a single small-bulge multi-shift QR sweep.

Purpose:



    CLAQR5 called by CLAQR0 performs a


    single small-bulge multi-shift QR sweep.

Parameters

WANTT 



          WANTT is LOGICAL


             WANTT = .true. if the triangular Schur factor


             is being computed.  WANTT is set to .false. otherwise.

WANTZ



          WANTZ is LOGICAL


             WANTZ = .true. if the unitary Schur factor is being


             computed.  WANTZ is set to .false. otherwise.

KACC22



          KACC22 is INTEGER with value 0, 1, or 2.


             Specifies the computation mode of far-from-diagonal


             orthogonal updates.


        = 0: CLAQR5 does not accumulate reflections and does not


             use matrix-matrix multiply to update far-from-diagonal


             matrix entries.


        = 1: CLAQR5 accumulates reflections and uses matrix-matrix


             multiply to update the far-from-diagonal matrix entries.


        = 2: Same as KACC22 = 1. This option used to enable exploiting


             the 2-by-2 structure during matrix multiplications, but


             this is no longer supported.

N



          N is INTEGER


             N is the order of the Hessenberg matrix H upon which this


             subroutine operates.

KTOP



          KTOP is INTEGER

KBOT



          KBOT is INTEGER


             These are the first and last rows and columns of an


             isolated diagonal block upon which the QR sweep is to be


             applied. It is assumed without a check that


                       either KTOP = 1  or   H(KTOP,KTOP-1) = 0


             and


                       either KBOT = N  or   H(KBOT+1,KBOT) = 0.

NSHFTS



          NSHFTS is INTEGER


             NSHFTS gives the number of simultaneous shifts.  NSHFTS


             must be positive and even.

S



          S is COMPLEX array, dimension (NSHFTS)


             S contains the shifts of origin that define the multi-


             shift QR sweep.  On output S may be reordered.

H



          H is COMPLEX array, dimension (LDH,N)


             On input H contains a Hessenberg matrix.  On output a


             multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied


             to the isolated diagonal block in rows and columns KTOP


             through KBOT.

LDH



          LDH is INTEGER


             LDH is the leading dimension of H just as declared in the


             calling procedure.  LDH >= MAX(1,N).

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 <= IHIZ <= N

Z



          Z is COMPLEX array, dimension (LDZ,IHIZ)


             If WANTZ = .TRUE., then the QR Sweep unitary


             similarity transformation is accumulated into


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


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

LDZ



          LDZ is INTEGER


             LDA is the leading dimension of Z just as declared in


             the calling procedure. LDZ >= N.

V



          V is COMPLEX array, dimension (LDV,NSHFTS/2)

LDV



          LDV is INTEGER


             LDV is the leading dimension of V as declared in the


             calling procedure.  LDV >= 3.

U



          U is COMPLEX array, dimension (LDU,2*NSHFTS)

LDU



          LDU is INTEGER


             LDU is the leading dimension of U just as declared in the


             in the calling subroutine.  LDU >= 2*NSHFTS.

NV



          NV is INTEGER


             NV is the number of rows in WV agailable for workspace.


             NV >= 1.

WV



          WV is COMPLEX array, dimension (LDWV,2*NSHFTS)

LDWV



          LDWV is INTEGER


             LDWV is the leading dimension of WV as declared in the


             in the calling subroutine.  LDWV >= NV.

NH



          NH is INTEGER


             NH is the number of columns in array WH available for


             workspace. NH >= 1.

WH



          WH is COMPLEX array, dimension (LDWH,NH)

LDWH



          LDWH is INTEGER


             Leading dimension of WH just as declared in the


             calling procedure.  LDWH >= 2*NSHFTS.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

Lars Karlsson, Daniel Kressner, and Bruno Lang

Thijs Steel, Department of Computer science, KU Leuven, Belgium

References:

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002.

Lars Karlsson, Daniel Kressner, and Bruno Lang, Optimally packed chains of bulges in multishift QR algorithms. ACM Trans. Math. Softw. 40, 2, Article 12 (February 2014).

subroutine dlaqr5 (logical wantt, logical wantz, integer kacc22, integer n, integer ktop, integer kbot, integer nshfts, double precision, dimension( * ) sr, double precision, dimension( * ) si, double precision, dimension( ldh, * ) h, integer ldh, integer iloz, integer ihiz, double precision, dimension( ldz, * ) z, integer ldz, double precision, dimension( ldv, * ) v, integer ldv, double precision, dimension( ldu, * ) u, integer ldu, integer nv, double precision, dimension( ldwv, * ) wv, integer ldwv, integer nh, double precision, dimension( ldwh, * ) wh, integer ldwh)

DLAQR5 performs a single small-bulge multi-shift QR sweep.

Purpose:



    DLAQR5, called by DLAQR0, performs a


    single small-bulge multi-shift QR sweep.

Parameters

WANTT 



          WANTT is LOGICAL


             WANTT = .true. if the quasi-triangular Schur factor


             is being computed.  WANTT is set to .false. otherwise.

WANTZ



          WANTZ is LOGICAL


             WANTZ = .true. if the orthogonal Schur factor is being


             computed.  WANTZ is set to .false. otherwise.

KACC22



          KACC22 is INTEGER with value 0, 1, or 2.


             Specifies the computation mode of far-from-diagonal


             orthogonal updates.


        = 0: DLAQR5 does not accumulate reflections and does not


             use matrix-matrix multiply to update far-from-diagonal


             matrix entries.


        = 1: DLAQR5 accumulates reflections and uses matrix-matrix


             multiply to update the far-from-diagonal matrix entries.


        = 2: Same as KACC22 = 1. This option used to enable exploiting


             the 2-by-2 structure during matrix multiplications, but


             this is no longer supported.

N



          N is INTEGER


             N is the order of the Hessenberg matrix H upon which this


             subroutine operates.

KTOP



          KTOP is INTEGER

KBOT



          KBOT is INTEGER


             These are the first and last rows and columns of an


             isolated diagonal block upon which the QR sweep is to be


             applied. It is assumed without a check that


                       either KTOP = 1  or   H(KTOP,KTOP-1) = 0


             and


                       either KBOT = N  or   H(KBOT+1,KBOT) = 0.

NSHFTS



          NSHFTS is INTEGER


             NSHFTS gives the number of simultaneous shifts.  NSHFTS


             must be positive and even.

SR



          SR is DOUBLE PRECISION array, dimension (NSHFTS)

SI



          SI is DOUBLE PRECISION array, dimension (NSHFTS)


             SR contains the real parts and SI contains the imaginary


             parts of the NSHFTS shifts of origin that define the


             multi-shift QR sweep.  On output SR and SI may be


             reordered.

H



          H is DOUBLE PRECISION array, dimension (LDH,N)


             On input H contains a Hessenberg matrix.  On output a


             multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied


             to the isolated diagonal block in rows and columns KTOP


             through KBOT.

LDH



          LDH is INTEGER


             LDH is the leading dimension of H just as declared in the


             calling procedure.  LDH >= MAX(1,N).

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 <= IHIZ <= N

Z



          Z is DOUBLE PRECISION array, dimension (LDZ,IHIZ)


             If WANTZ = .TRUE., then the QR Sweep orthogonal


             similarity transformation is accumulated into


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


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

LDZ



          LDZ is INTEGER


             LDA is the leading dimension of Z just as declared in


             the calling procedure. LDZ >= N.

V



          V is DOUBLE PRECISION array, dimension (LDV,NSHFTS/2)

LDV



          LDV is INTEGER


             LDV is the leading dimension of V as declared in the


             calling procedure.  LDV >= 3.

U



          U is DOUBLE PRECISION array, dimension (LDU,2*NSHFTS)

LDU



          LDU is INTEGER


             LDU is the leading dimension of U just as declared in the


             in the calling subroutine.  LDU >= 2*NSHFTS.

NV



          NV is INTEGER


             NV is the number of rows in WV agailable for workspace.


             NV >= 1.

WV



          WV is DOUBLE PRECISION array, dimension (LDWV,2*NSHFTS)

LDWV



          LDWV is INTEGER


             LDWV is the leading dimension of WV as declared in the


             in the calling subroutine.  LDWV >= NV.

NH



          NH is INTEGER


             NH is the number of columns in array WH available for


             workspace. NH >= 1.

WH



          WH is DOUBLE PRECISION array, dimension (LDWH,NH)

LDWH



          LDWH is INTEGER


             Leading dimension of WH just as declared in the


             calling procedure.  LDWH >= 2*NSHFTS.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

Lars Karlsson, Daniel Kressner, and Bruno Lang

Thijs Steel, Department of Computer science, KU Leuven, Belgium

References:

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002.

Lars Karlsson, Daniel Kressner, and Bruno Lang, Optimally packed chains of bulges in multishift QR algorithms. ACM Trans. Math. Softw. 40, 2, Article 12 (February 2014).

subroutine slaqr5 (logical wantt, logical wantz, integer kacc22, integer n, integer ktop, integer kbot, integer nshfts, real, dimension( * ) sr, real, dimension( * ) si, real, dimension( ldh, * ) h, integer ldh, integer iloz, integer ihiz, real, dimension( ldz, * ) z, integer ldz, real, dimension( ldv, * ) v, integer ldv, real, dimension( ldu, * ) u, integer ldu, integer nv, real, dimension( ldwv, * ) wv, integer ldwv, integer nh, real, dimension( ldwh, * ) wh, integer ldwh)

SLAQR5 performs a single small-bulge multi-shift QR sweep.

Purpose:



    SLAQR5, called by SLAQR0, performs a


    single small-bulge multi-shift QR sweep.

Parameters

WANTT 



          WANTT is LOGICAL


             WANTT = .true. if the quasi-triangular Schur factor


             is being computed.  WANTT is set to .false. otherwise.

WANTZ



          WANTZ is LOGICAL


             WANTZ = .true. if the orthogonal Schur factor is being


             computed.  WANTZ is set to .false. otherwise.

KACC22



          KACC22 is INTEGER with value 0, 1, or 2.


             Specifies the computation mode of far-from-diagonal


             orthogonal updates.


        = 0: SLAQR5 does not accumulate reflections and does not


             use matrix-matrix multiply to update far-from-diagonal


             matrix entries.


        = 1: SLAQR5 accumulates reflections and uses matrix-matrix


             multiply to update the far-from-diagonal matrix entries.


        = 2: Same as KACC22 = 1. This option used to enable exploiting


             the 2-by-2 structure during matrix multiplications, but


             this is no longer supported.

N



          N is INTEGER


             N is the order of the Hessenberg matrix H upon which this


             subroutine operates.

KTOP



          KTOP is INTEGER

KBOT



          KBOT is INTEGER


             These are the first and last rows and columns of an


             isolated diagonal block upon which the QR sweep is to be


             applied. It is assumed without a check that


                       either KTOP = 1  or   H(KTOP,KTOP-1) = 0


             and


                       either KBOT = N  or   H(KBOT+1,KBOT) = 0.

NSHFTS



          NSHFTS is INTEGER


             NSHFTS gives the number of simultaneous shifts.  NSHFTS


             must be positive and even.

SR



          SR is REAL array, dimension (NSHFTS)

SI



          SI is REAL array, dimension (NSHFTS)


             SR contains the real parts and SI contains the imaginary


             parts of the NSHFTS shifts of origin that define the


             multi-shift QR sweep.  On output SR and SI may be


             reordered.

H



          H is REAL array, dimension (LDH,N)


             On input H contains a Hessenberg matrix.  On output a


             multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied


             to the isolated diagonal block in rows and columns KTOP


             through KBOT.

LDH



          LDH is INTEGER


             LDH is the leading dimension of H just as declared in the


             calling procedure.  LDH >= MAX(1,N).

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 <= IHIZ <= N

Z



          Z is REAL array, dimension (LDZ,IHIZ)


             If WANTZ = .TRUE., then the QR Sweep orthogonal


             similarity transformation is accumulated into


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


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

LDZ



          LDZ is INTEGER


             LDA is the leading dimension of Z just as declared in


             the calling procedure. LDZ >= N.

V



          V is REAL array, dimension (LDV,NSHFTS/2)

LDV



          LDV is INTEGER


             LDV is the leading dimension of V as declared in the


             calling procedure.  LDV >= 3.

U



          U is REAL array, dimension (LDU,2*NSHFTS)

LDU



          LDU is INTEGER


             LDU is the leading dimension of U just as declared in the


             in the calling subroutine.  LDU >= 2*NSHFTS.

NV



          NV is INTEGER


             NV is the number of rows in WV agailable for workspace.


             NV >= 1.

WV



          WV is REAL array, dimension (LDWV,2*NSHFTS)

LDWV



          LDWV is INTEGER


             LDWV is the leading dimension of WV as declared in the


             in the calling subroutine.  LDWV >= NV.

NH



          NH is INTEGER


             NH is the number of columns in array WH available for


             workspace. NH >= 1.

WH



          WH is REAL array, dimension (LDWH,NH)

LDWH



          LDWH is INTEGER


             Leading dimension of WH just as declared in the


             calling procedure.  LDWH >= 2*NSHFTS.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

Lars Karlsson, Daniel Kressner, and Bruno Lang

Thijs Steel, Department of Computer science, KU Leuven, Belgium

References:

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002.

Lars Karlsson, Daniel Kressner, and Bruno Lang, Optimally packed chains of bulges in multishift QR algorithms. ACM Trans. Math. Softw. 40, 2, Article 12 (February 2014).

subroutine zlaqr5 (logical wantt, logical wantz, integer kacc22, integer n, integer ktop, integer kbot, integer nshfts, complex*16, dimension( * ) s, complex*16, dimension( ldh, * ) h, integer ldh, integer iloz, integer ihiz, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( ldv, * ) v, integer ldv, complex*16, dimension( ldu, * ) u, integer ldu, integer nv, complex*16, dimension( ldwv, * ) wv, integer ldwv, integer nh, complex*16, dimension( ldwh, * ) wh, integer ldwh)

ZLAQR5 performs a single small-bulge multi-shift QR sweep.

Purpose:



    ZLAQR5, called by ZLAQR0, performs a


    single small-bulge multi-shift QR sweep.

Parameters

WANTT 



          WANTT is LOGICAL


             WANTT = .true. if the triangular Schur factor


             is being computed.  WANTT is set to .false. otherwise.

WANTZ



          WANTZ is LOGICAL


             WANTZ = .true. if the unitary Schur factor is being


             computed.  WANTZ is set to .false. otherwise.

KACC22



          KACC22 is INTEGER with value 0, 1, or 2.


             Specifies the computation mode of far-from-diagonal


             orthogonal updates.


        = 0: ZLAQR5 does not accumulate reflections and does not


             use matrix-matrix multiply to update far-from-diagonal


             matrix entries.


        = 1: ZLAQR5 accumulates reflections and uses matrix-matrix


             multiply to update the far-from-diagonal matrix entries.


        = 2: Same as KACC22 = 1. This option used to enable exploiting


             the 2-by-2 structure during matrix multiplications, but


             this is no longer supported.

N



          N is INTEGER


             N is the order of the Hessenberg matrix H upon which this


             subroutine operates.

KTOP



          KTOP is INTEGER

KBOT



          KBOT is INTEGER


             These are the first and last rows and columns of an


             isolated diagonal block upon which the QR sweep is to be


             applied. It is assumed without a check that


                       either KTOP = 1  or   H(KTOP,KTOP-1) = 0


             and


                       either KBOT = N  or   H(KBOT+1,KBOT) = 0.

NSHFTS



          NSHFTS is INTEGER


             NSHFTS gives the number of simultaneous shifts.  NSHFTS


             must be positive and even.

S



          S is COMPLEX*16 array, dimension (NSHFTS)


             S contains the shifts of origin that define the multi-


             shift QR sweep.  On output S may be reordered.

H



          H is COMPLEX*16 array, dimension (LDH,N)


             On input H contains a Hessenberg matrix.  On output a


             multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied


             to the isolated diagonal block in rows and columns KTOP


             through KBOT.

LDH



          LDH is INTEGER


             LDH is the leading dimension of H just as declared in the


             calling procedure.  LDH >= MAX(1,N).

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 <= IHIZ <= N

Z



          Z is COMPLEX*16 array, dimension (LDZ,IHIZ)


             If WANTZ = .TRUE., then the QR Sweep unitary


             similarity transformation is accumulated into


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


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

LDZ



          LDZ is INTEGER


             LDA is the leading dimension of Z just as declared in


             the calling procedure. LDZ >= N.

V



          V is COMPLEX*16 array, dimension (LDV,NSHFTS/2)

LDV



          LDV is INTEGER


             LDV is the leading dimension of V as declared in the


             calling procedure.  LDV >= 3.

U



          U is COMPLEX*16 array, dimension (LDU,2*NSHFTS)

LDU



          LDU is INTEGER


             LDU is the leading dimension of U just as declared in the


             in the calling subroutine.  LDU >= 2*NSHFTS.

NV



          NV is INTEGER


             NV is the number of rows in WV agailable for workspace.


             NV >= 1.

WV



          WV is COMPLEX*16 array, dimension (LDWV,2*NSHFTS)

LDWV



          LDWV is INTEGER


             LDWV is the leading dimension of WV as declared in the


             in the calling subroutine.  LDWV >= NV.

NH



          NH is INTEGER


             NH is the number of columns in array WH available for


             workspace. NH >= 1.

WH



          WH is COMPLEX*16 array, dimension (LDWH,NH)

LDWH



          LDWH is INTEGER


             Leading dimension of WH just as declared in the


             calling procedure.  LDWH >= 2*NSHFTS.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

Lars Karlsson, Daniel Kressner, and Bruno Lang

Thijs Steel, Department of Computer science, KU Leuven, Belgium

References:

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002.

Lars Karlsson, Daniel Kressner, and Bruno Lang, Optimally packed chains of bulges in multishift QR algorithms. ACM Trans. Math. Softw. 40, 2, Article 12 (February 2014).

Author

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