laqr5 - laqr5: step in hseqr

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subroutineclaqr5(logicalwantt,logicalwantz,integerkacc22,integern,integerktop,integerkbot,integernshfts,complex,dimension(*)s,complex,dimension(ldh,*)h,integerldh,integeriloz,integerihiz,complex,dimension(ldz,*)z,integerldz,complex,dimension(ldv,*)v,integerldv,complex,dimension(ldu,*)u,integerldu,integernv,complex,dimension(ldwv,*)wv,integerldwv,integernh,complex,dimension(ldwh,*)wh,integerldwh)CLAQR5 performs a single small-bulge multi-shift QR sweep. Purpose: CLAQR5 called by CLAQR0 performs a single small-bulge multi-shift QR sweep. ParametersWANTT 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). subroutinedlaqr5(logicalwantt,logicalwantz,integerkacc22,integern,integerktop,integerkbot,integernshfts,doubleprecision,dimension(*)sr,doubleprecision,dimension(*)si,doubleprecision,dimension(ldh,*)h,integerldh,integeriloz,integerihiz,doubleprecision,dimension(ldz,*)z,integerldz,doubleprecision,dimension(ldv,*)v,integerldv,doubleprecision,dimension(ldu,*)u,integerldu,integernv,doubleprecision,dimension(ldwv,*)wv,integerldwv,integernh,doubleprecision,dimension(ldwh,*)wh,integerldwh)DLAQR5 performs a single small-bulge multi-shift QR sweep. Purpose: DLAQR5, called by DLAQR0, performs a single small-bulge multi-shift QR sweep. ParametersWANTT 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). subroutineslaqr5(logicalwantt,logicalwantz,integerkacc22,integern,integerktop,integerkbot,integernshfts,real,dimension(*)sr,real,dimension(*)si,real,dimension(ldh,*)h,integerldh,integeriloz,integerihiz,real,dimension(ldz,*)z,integerldz,real,dimension(ldv,*)v,integerldv,real,dimension(ldu,*)u,integerldu,integernv,real,dimension(ldwv,*)wv,integerldwv,integernh,real,dimension(ldwh,*)wh,integerldwh)SLAQR5 performs a single small-bulge multi-shift QR sweep. Purpose: SLAQR5, called by SLAQR0, performs a single small-bulge multi-shift QR sweep. ParametersWANTT 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). subroutinezlaqr5(logicalwantt,logicalwantz,integerkacc22,integern,integerktop,integerkbot,integernshfts,complex*16,dimension(*)s,complex*16,dimension(ldh,*)h,integerldh,integeriloz,integerihiz,complex*16,dimension(ldz,*)z,integerldz,complex*16,dimension(ldv,*)v,integerldv,complex*16,dimension(ldu,*)u,integerldu,integernv,complex*16,dimension(ldwv,*)wv,integerldwv,integernh,complex*16,dimension(ldwh,*)wh,integerldwh)ZLAQR5 performs a single small-bulge multi-shift QR sweep. Purpose: ZLAQR5, called by ZLAQR0, performs a single small-bulge multi-shift QR sweep. ParametersWANTT 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).

laqr5 - laqr5: step in hseqr

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.