--- name: "lahqr" 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/lahqr" --- # lahqr(3) lahqr: eig of Hessenberg, step in hseqr ## 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). !> ``` Definition at line **193** of file **clahqr.f**. ### 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). !> ``` Definition at line **205** of file **dlahqr.f**. ### 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). !> ``` Definition at line **205** of file **slahqr.f**. ### 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). !> ``` Definition at line **193** of file **zlahqr.f**. ## Author Generated automatically by Doxygen for LAPACK from the source code.