laqr2(3) Library Functions Manual laqr2(3) NAME laqr2 - laqr2: step in hseqr SYNOPSIS Functions subroutine claqr2 (wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz, ns, nd, sh, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork) CLAQR2 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). subroutine dlaqr2 (wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz, ns, nd, sr, si, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork) DLAQR2 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). subroutine slaqr2 (wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz, ns, nd, sr, si, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork) SLAQR2 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). subroutine zlaqr2 (wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz, ns, nd, sh, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork) ZLAQR2 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). Detailed Description Function Documentation subroutine claqr2 (logical wantt, logical wantz, integer n, integer ktop, integer kbot, integer nw, complex, dimension( ldh, * ) h, integer ldh, integer iloz, integer ihiz, complex, dimension( ldz, * ) z, integer ldz, integer ns, integer nd, complex, dimension( * ) sh, complex, dimension( ldv, * ) v, integer ldv, integer nh, complex, dimension( ldt, * ) t, integer ldt, integer nv, complex, dimension( ldwv, * ) wv, integer ldwv, complex, dimension( * ) work, integer lwork) CLAQR2 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). Purpose: CLAQR2 is identical to CLAQR3 except that it avoids recursion by calling CLAHQR instead of CLAQR4. Aggressive early deflation: This subroutine accepts as input an upper Hessenberg matrix H and performs an unitary 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 unitary similarity transformation of H. It is to be hoped that the final version of H has many zero subdiagonal entries. Parameters WANTT WANTT is LOGICAL If .TRUE., then the Hessenberg matrix H is fully updated so that the 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 unitary matrix Z is updated so so that the unitary 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 unitary 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 <= NW <= (KBOT-KTOP+1). H H is COMPLEX 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 a unitary 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 <= 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 <= ILOZ <= IHIZ <= N. Z Z is COMPLEX array, dimension (LDZ,N) IF WANTZ is .TRUE., then on output, the unitary 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 <= 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. SH SH is COMPLEX array, dimension (KBOT) On output, approximate eigenvalues that may be used for shifts are stored in SH(KBOT-ND-NS+1) through SR(KBOT-ND). Converged eigenvalues are stored in SH(KBOT-ND+1) through SH(KBOT). V V is COMPLEX 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 <= LDV NH NH is INTEGER The number of columns of T. NH >= NW. T T is COMPLEX array, dimension (LDT,NW) LDT LDT is INTEGER The leading dimension of T just as declared in the calling subroutine. NW <= LDT NV NV is INTEGER The number of rows of work array WV available for workspace. NV >= NW. WV WV is COMPLEX array, dimension (LDWV,NW) LDWV LDWV is INTEGER The leading dimension of W just as declared in the calling subroutine. NW <= LDV WORK WORK is COMPLEX 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; CLAQR2 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. 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 Definition at line 266 of file claqr2.f. subroutine dlaqr2 (logical wantt, logical wantz, integer n, integer ktop, integer kbot, integer nw, double precision, dimension( ldh, * ) h, integer ldh, integer iloz, integer ihiz, double precision, dimension( ldz, * ) z, integer ldz, integer ns, integer nd, double precision, dimension( * ) sr, double precision, dimension( * ) si, double precision, dimension( ldv, * ) v, integer ldv, integer nh, double precision, dimension( ldt, * ) t, integer ldt, integer nv, double precision, dimension( ldwv, * ) wv, integer ldwv, double precision, dimension( * ) work, integer lwork) DLAQR2 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). Purpose: DLAQR2 is identical to DLAQR3 except that it avoids recursion by calling DLAHQR instead of DLAQR4. Aggressive early deflation: This subroutine 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. Parameters 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 <= NW <= (KBOT-KTOP+1). H H is DOUBLE PRECISION 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 <= 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 <= ILOZ <= IHIZ <= N. Z Z is DOUBLE PRECISION 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 <= 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 DOUBLE PRECISION array, dimension (KBOT) SI SI is DOUBLE PRECISION 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 DOUBLE PRECISION 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 <= LDV NH NH is INTEGER The number of columns of T. NH >= NW. T T is DOUBLE PRECISION array, dimension (LDT,NW) LDT LDT is INTEGER The leading dimension of T just as declared in the calling subroutine. NW <= LDT NV NV is INTEGER The number of rows of work array WV available for workspace. NV >= NW. WV WV is DOUBLE PRECISION array, dimension (LDWV,NW) LDWV LDWV is INTEGER The leading dimension of W just as declared in the calling subroutine. NW <= LDV WORK WORK is DOUBLE PRECISION 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; DLAQR2 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. 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 Definition at line 275 of file dlaqr2.f. subroutine slaqr2 (logical wantt, logical wantz, integer n, integer ktop, integer kbot, integer nw, real, dimension( ldh, * ) h, integer ldh, integer iloz, integer ihiz, real, dimension( ldz, * ) z, integer ldz, integer ns, integer nd, real, dimension( * ) sr, real, dimension( * ) si, real, dimension( ldv, * ) v, integer ldv, integer nh, real, dimension( ldt, * ) t, integer ldt, integer nv, real, dimension( ldwv, * ) wv, integer ldwv, real, dimension( * ) work, integer lwork) SLAQR2 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). Purpose: SLAQR2 is identical to SLAQR3 except that it avoids recursion by calling SLAHQR instead of SLAQR4. Aggressive early deflation: This subroutine 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. Parameters 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 <= NW <= (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 <= 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 <= ILOZ <= IHIZ <= 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 <= 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 <= LDV NH NH is INTEGER The number of columns of T. NH >= 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 <= LDT NV NV is INTEGER The number of rows of work array WV available for workspace. NV >= 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 <= 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; SLAQR2 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. 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 Definition at line 275 of file slaqr2.f. subroutine zlaqr2 (logical wantt, logical wantz, integer n, integer ktop, integer kbot, integer nw, complex*16, dimension( ldh, * ) h, integer ldh, integer iloz, integer ihiz, complex*16, dimension( ldz, * ) z, integer ldz, integer ns, integer nd, complex*16, dimension( * ) sh, complex*16, dimension( ldv, * ) v, integer ldv, integer nh, complex*16, dimension( ldt, * ) t, integer ldt, integer nv, complex*16, dimension( ldwv, * ) wv, integer ldwv, complex*16, dimension( * ) work, integer lwork) ZLAQR2 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). Purpose: ZLAQR2 is identical to ZLAQR3 except that it avoids recursion by calling ZLAHQR instead of ZLAQR4. Aggressive early deflation: ZLAQR2 accepts as input an upper Hessenberg matrix H and performs an unitary 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 unitary similarity transformation of H. It is to be hoped that the final version of H has many zero subdiagonal entries. Parameters WANTT WANTT is LOGICAL If .TRUE., then the Hessenberg matrix H is fully updated so that the 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 unitary matrix Z is updated so so that the unitary 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 unitary 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 <= NW <= (KBOT-KTOP+1). H H is COMPLEX*16 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 a unitary 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 <= 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 <= ILOZ <= IHIZ <= N. Z Z is COMPLEX*16 array, dimension (LDZ,N) IF WANTZ is .TRUE., then on output, the unitary 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 <= 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. SH SH is COMPLEX*16 array, dimension (KBOT) On output, approximate eigenvalues that may be used for shifts are stored in SH(KBOT-ND-NS+1) through SR(KBOT-ND). Converged eigenvalues are stored in SH(KBOT-ND+1) through SH(KBOT). V V is COMPLEX*16 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 <= LDV NH NH is INTEGER The number of columns of T. NH >= NW. T T is COMPLEX*16 array, dimension (LDT,NW) LDT LDT is INTEGER The leading dimension of T just as declared in the calling subroutine. NW <= LDT NV NV is INTEGER The number of rows of work array WV available for workspace. NV >= NW. WV WV is COMPLEX*16 array, dimension (LDWV,NW) LDWV LDWV is INTEGER The leading dimension of W just as declared in the calling subroutine. NW <= LDV WORK WORK is COMPLEX*16 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; ZLAQR2 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. 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 Definition at line 267 of file zlaqr2.f. Author Generated automatically by Doxygen for LAPACK from the source code. LAPACK Version 3.12.0 laqr2(3)