--- name: "laqr5" section: "3" platform: "linux" package: "lapack-doc" version: "3.12.1-2" release: "Arch Linux 2026-11-01" platforms: ["linux"] url: "https://new.carta.tech/laqr5" --- # laqr5(3) laqr5: step in hseqr ## 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). Definition at line **254** of file **claqr5.f**. ### 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). Definition at line **262** of file **dlaqr5.f**. ### 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). Definition at line **262** of file **slaqr5.f**. ### 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). Definition at line **254** of file **zlaqr5.f**. ## Author Generated automatically by Doxygen for LAPACK from the source code.