--- name: "laqz3" 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/laqz3" --- # laqz3(3) laqz3: step in ggev3, gges3 ## NAME laqz3 - laqz3: step in ggev3, gges3 ## SYNOPSIS ### Functions subroutine **claqz3** (ilschur, ilq, ilz, n, ilo, ihi, nshifts, nblock\_desired, alpha, beta, a, lda, b, ldb, q, ldq, z, ldz, qc, ldqc, zc, ldzc, work, lwork, info)\ **CLAQZ3** recursive subroutine **dlaqz3** (ilschur, ilq, ilz, n, ilo, ihi, nw, a, lda, b, ldb, q, ldq, z, ldz, ns, nd, alphar, alphai, beta, qc, ldqc, zc, ldzc, work, lwork, rec, info)\ **DLAQZ3** recursive subroutine **slaqz3** (ilschur, ilq, ilz, n, ilo, ihi, nw, a, lda, b, ldb, q, ldq, z, ldz, ns, nd, alphar, alphai, beta, qc, ldqc, zc, ldzc, work, lwork, rec, info)\ **SLAQZ3** subroutine **zlaqz3** (ilschur, ilq, ilz, n, ilo, ihi, nshifts, nblock\_desired, alpha, beta, a, lda, b, ldb, q, ldq, z, ldz, qc, ldqc, zc, ldzc, work, lwork, info)\ **ZLAQZ3** ## Detailed Description ## Function Documentation ### subroutine claqz3 (logical, intent(in) ilschur, logical, intent(in) ilq, logical, intent(in) ilz, integer, intent(in) n, integer, intent(in) ilo, integer, intent(in) ihi, integer, intent(in) nshifts, integer, intent(in) nblock\_desired, complex, dimension( \* ), intent(inout) alpha, complex, dimension( \* ), intent(inout) beta, complex, dimension( lda, \* ), intent(inout) a, integer, intent(in) lda, complex, dimension( ldb, \* ), intent(inout) b, integer, intent(in) ldb, complex, dimension( ldq, \* ), intent(inout) q, integer, intent(in) ldq, complex, dimension( ldz, \* ), intent(inout) z, integer, intent(in) ldz, complex, dimension( ldqc, \* ), intent(inout) qc, integer, intent(in) ldqc, complex, dimension( ldzc, \* ), intent(inout) zc, integer, intent(in) ldzc, complex, dimension( \* ), intent(inout) work, integer, intent(in) lwork, integer, intent(out) info) **CLAQZ3** **Purpose:** ``` !> !> CLAQZ3 Executes a single multishift QZ sweep !> ``` **Parameters** *ILSCHUR* ``` !> ILSCHUR is LOGICAL !> Determines whether or not to update the full Schur form !> ``` *ILQ* ``` !> ILQ is LOGICAL !> Determines whether or not to update the matrix Q !> ``` *ILZ* ``` !> ILZ is LOGICAL !> Determines whether or not to update the matrix Z !> ``` *N* ``` !> N is INTEGER !> The order of the matrices A, B, Q, and Z. N >= 0. !> ``` *ILO* ``` !> ILO is INTEGER !> ``` *IHI* ``` !> IHI is INTEGER !> ``` *NSHIFTS* ``` !> NSHIFTS is INTEGER !> The desired number of shifts to use !> ``` *NBLOCK\_DESIRED* ``` !> NBLOCK_DESIRED is INTEGER !> The desired size of the computational windows !> ``` *ALPHA* ``` !> ALPHA is COMPLEX array. SR contains !> the alpha parts of the shifts to use. !> ``` *BETA* ``` !> BETA is COMPLEX array. SS contains !> the scale of the shifts to use. !> ``` *A* ``` !> A is COMPLEX array, dimension (LDA, N) !> ``` *LDA* ``` !> LDA is INTEGER !> The leading dimension of the array A. LDA >= max( 1, N ). !> ``` *B* ``` !> B is COMPLEX array, dimension (LDB, N) !> ``` *LDB* ``` !> LDB is INTEGER !> The leading dimension of the array B. LDB >= max( 1, N ). !> ``` *Q* ``` !> Q is COMPLEX array, dimension (LDQ, N) !> ``` *LDQ* ``` !> LDQ is INTEGER !> ``` *Z* ``` !> Z is COMPLEX array, dimension (LDZ, N) !> ``` *LDZ* ``` !> LDZ is INTEGER !> ``` *QC* ``` !> QC is COMPLEX array, dimension (LDQC, NBLOCK_DESIRED) !> ``` *LDQC* ``` !> LDQC is INTEGER !> ``` *ZC* ``` !> ZC is COMPLEX array, dimension (LDZC, NBLOCK_DESIRED) !> ``` *LDZC* ``` !> LDZ is INTEGER !> ``` *WORK* ``` !> WORK is COMPLEX array, dimension (MAX(1,LWORK)) !> On exit, if INFO >= 0, WORK(1) returns the optimal LWORK. !> ``` *LWORK* ``` !> LWORK is INTEGER !> The dimension of the array WORK. LWORK >= max(1,N). !> !> If LWORK = -1, then a workspace query is assumed; the routine !> only calculates the optimal size of the WORK array, returns !> this value as the first entry of the WORK array, and no error !> message related to LWORK is issued by XERBLA. !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> ``` **Author** Thijs Steel, KU Leuven **Date** May 2020 Definition at line **203** of file **claqz3.f**. ### recursive subroutine dlaqz3 (logical, intent(in) ilschur, logical, intent(in) ilq, logical, intent(in) ilz, integer, intent(in) n, integer, intent(in) ilo, integer, intent(in) ihi, integer, intent(in) nw, double precision, dimension( lda, \* ), intent(inout) a, integer, intent(in) lda, double precision, dimension( ldb, \* ), intent(inout) b, integer, intent(in) ldb, double precision, dimension( ldq, \* ), intent(inout) q, integer, intent(in) ldq, double precision, dimension( ldz, \* ), intent(inout) z, integer, intent(in) ldz, integer, intent(out) ns, integer, intent(out) nd, double precision, dimension( \* ), intent(inout) alphar, double precision, dimension( \* ), intent(inout) alphai, double precision, dimension( \* ), intent(inout) beta, double precision, dimension( ldqc, \* ) qc, integer, intent(in) ldqc, double precision, dimension( ldzc, \* ) zc, integer, intent(in) ldzc, double precision, dimension( \* ) work, integer, intent(in) lwork, integer, intent(in) rec, integer, intent(out) info) **DLAQZ3** **Purpose:** ``` !> !> DLAQZ3 performs AED !> ``` **Parameters** *ILSCHUR* ``` !> ILSCHUR is LOGICAL !> Determines whether or not to update the full Schur form !> ``` *ILQ* ``` !> ILQ is LOGICAL !> Determines whether or not to update the matrix Q !> ``` *ILZ* ``` !> ILZ is LOGICAL !> Determines whether or not to update the matrix Z !> ``` *N* ``` !> N is INTEGER !> The order of the matrices A, B, Q, and Z. N >= 0. !> ``` *ILO* ``` !> ILO is INTEGER !> ``` *IHI* ``` !> IHI is INTEGER !> ILO and IHI mark the rows and columns of (A,B) which !> are to be normalized !> ``` *NW* ``` !> NW is INTEGER !> The desired size of the deflation window. !> ``` *A* ``` !> A is DOUBLE PRECISION array, dimension (LDA, N) !> ``` *LDA* ``` !> LDA is INTEGER !> The leading dimension of the array A. LDA >= max( 1, N ). !> ``` *B* ``` !> B is DOUBLE PRECISION array, dimension (LDB, N) !> ``` *LDB* ``` !> LDB is INTEGER !> The leading dimension of the array B. LDB >= max( 1, N ). !> ``` *Q* ``` !> Q is DOUBLE PRECISION array, dimension (LDQ, N) !> ``` *LDQ* ``` !> LDQ is INTEGER !> ``` *Z* ``` !> Z is DOUBLE PRECISION array, dimension (LDZ, N) !> ``` *LDZ* ``` !> LDZ is INTEGER !> ``` *NS* ``` !> NS is INTEGER !> The number of unconverged eigenvalues available to !> use as shifts. !> ``` *ND* ``` !> ND is INTEGER !> The number of converged eigenvalues found. !> ``` *ALPHAR* ``` !> ALPHAR is DOUBLE PRECISION array, dimension (N) !> The real parts of each scalar alpha defining an eigenvalue !> of GNEP. !> ``` *ALPHAI* ``` !> ALPHAI is DOUBLE PRECISION array, dimension (N) !> The imaginary parts of each scalar alpha defining an !> eigenvalue of GNEP. !> If ALPHAI(j) is zero, then the j-th eigenvalue is real; if !> positive, then the j-th and (j+1)-st eigenvalues are a !> complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j). !> ``` *BETA* ``` !> BETA is DOUBLE PRECISION array, dimension (N) !> The scalars beta that define the eigenvalues of GNEP. !> Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and !> beta = BETA(j) represent the j-th eigenvalue of the matrix !> pair (A,B), in one of the forms lambda = alpha/beta or !> mu = beta/alpha. Since either lambda or mu may overflow, !> they should not, in general, be computed. !> ``` *QC* ``` !> QC is DOUBLE PRECISION array, dimension (LDQC, NW) !> ``` *LDQC* ``` !> LDQC is INTEGER !> ``` *ZC* ``` !> ZC is DOUBLE PRECISION array, dimension (LDZC, NW) !> ``` *LDZC* ``` !> LDZ is INTEGER !> ``` *WORK* ``` !> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) !> On exit, if INFO >= 0, WORK(1) returns the optimal LWORK. !> ``` *LWORK* ``` !> LWORK is INTEGER !> The dimension of the array WORK. LWORK >= max(1,N). !> !> If LWORK = -1, then a workspace query is assumed; the routine !> only calculates the optimal size of the WORK array, returns !> this value as the first entry of the WORK array, and no error !> message related to LWORK is issued by XERBLA. !> ``` *REC* ``` !> REC is INTEGER !> REC indicates the current recursion level. Should be set !> to 0 on first call. !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> ``` **Author** Thijs Steel, KU Leuven **Date** May 2020 Definition at line **235** of file **dlaqz3.f**. ### recursive subroutine slaqz3 (logical, intent(in) ilschur, logical, intent(in) ilq, logical, intent(in) ilz, integer, intent(in) n, integer, intent(in) ilo, integer, intent(in) ihi, integer, intent(in) nw, real, dimension( lda, \* ), intent(inout) a, integer, intent(in) lda, real, dimension( ldb, \* ), intent(inout) b, integer, intent(in) ldb, real, dimension( ldq, \* ), intent(inout) q, integer, intent(in) ldq, real, dimension( ldz, \* ), intent(inout) z, integer, intent(in) ldz, integer, intent(out) ns, integer, intent(out) nd, real, dimension( \* ), intent(inout) alphar, real, dimension( \* ), intent(inout) alphai, real, dimension( \* ), intent(inout) beta, real, dimension( ldqc, \* ) qc, integer, intent(in) ldqc, real, dimension( ldzc, \* ) zc, integer, intent(in) ldzc, real, dimension( \* ) work, integer, intent(in) lwork, integer, intent(in) rec, integer, intent(out) info) **SLAQZ3** **Purpose:** ``` !> !> SLAQZ3 performs AED !> ``` **Parameters** *ILSCHUR* ``` !> ILSCHUR is LOGICAL !> Determines whether or not to update the full Schur form !> ``` *ILQ* ``` !> ILQ is LOGICAL !> Determines whether or not to update the matrix Q !> ``` *ILZ* ``` !> ILZ is LOGICAL !> Determines whether or not to update the matrix Z !> ``` *N* ``` !> N is INTEGER !> The order of the matrices A, B, Q, and Z. N >= 0. !> ``` *ILO* ``` !> ILO is INTEGER !> ``` *IHI* ``` !> IHI is INTEGER !> ILO and IHI mark the rows and columns of (A,B) which !> are to be normalized !> ``` *NW* ``` !> NW is INTEGER !> The desired size of the deflation window. !> ``` *A* ``` !> A is REAL array, dimension (LDA, N) !> ``` *LDA* ``` !> LDA is INTEGER !> The leading dimension of the array A. LDA >= max( 1, N ). !> ``` *B* ``` !> B is REAL array, dimension (LDB, N) !> ``` *LDB* ``` !> LDB is INTEGER !> The leading dimension of the array B. LDB >= max( 1, N ). !> ``` *Q* ``` !> Q is REAL array, dimension (LDQ, N) !> ``` *LDQ* ``` !> LDQ is INTEGER !> ``` *Z* ``` !> Z is REAL array, dimension (LDZ, N) !> ``` *LDZ* ``` !> LDZ is INTEGER !> ``` *NS* ``` !> NS is INTEGER !> The number of unconverged eigenvalues available to !> use as shifts. !> ``` *ND* ``` !> ND is INTEGER !> The number of converged eigenvalues found. !> ``` *ALPHAR* ``` !> ALPHAR is REAL array, dimension (N) !> The real parts of each scalar alpha defining an eigenvalue !> of GNEP. !> ``` *ALPHAI* ``` !> ALPHAI is REAL array, dimension (N) !> The imaginary parts of each scalar alpha defining an !> eigenvalue of GNEP. !> If ALPHAI(j) is zero, then the j-th eigenvalue is real; if !> positive, then the j-th and (j+1)-st eigenvalues are a !> complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j). !> ``` *BETA* ``` !> BETA is REAL array, dimension (N) !> The scalars beta that define the eigenvalues of GNEP. !> Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and !> beta = BETA(j) represent the j-th eigenvalue of the matrix !> pair (A,B), in one of the forms lambda = alpha/beta or !> mu = beta/alpha. Since either lambda or mu may overflow, !> they should not, in general, be computed. !> ``` *QC* ``` !> QC is REAL array, dimension (LDQC, NW) !> ``` *LDQC* ``` !> LDQC is INTEGER !> ``` *ZC* ``` !> ZC is REAL array, dimension (LDZC, NW) !> ``` *LDZC* ``` !> LDZ is INTEGER !> ``` *WORK* ``` !> WORK is REAL array, dimension (MAX(1,LWORK)) !> On exit, if INFO >= 0, WORK(1) returns the optimal LWORK. !> ``` *LWORK* ``` !> LWORK is INTEGER !> The dimension of the array WORK. LWORK >= max(1,N). !> !> If LWORK = -1, then a workspace query is assumed; the routine !> only calculates the optimal size of the WORK array, returns !> this value as the first entry of the WORK array, and no error !> message related to LWORK is issued by XERBLA. !> ``` *REC* ``` !> REC is INTEGER !> REC indicates the current recursion level. Should be set !> to 0 on first call. !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> ``` **Author** Thijs Steel, KU Leuven **Date** May 2020 Definition at line **234** of file **slaqz3.f**. ### subroutine zlaqz3 (logical, intent(in) ilschur, logical, intent(in) ilq, logical, intent(in) ilz, integer, intent(in) n, integer, intent(in) ilo, integer, intent(in) ihi, integer, intent(in) nshifts, integer, intent(in) nblock\_desired, complex\*16, dimension( \* ), intent(inout) alpha, complex\*16, dimension( \* ), intent(inout) beta, complex\*16, dimension( lda, \* ), intent(inout) a, integer, intent(in) lda, complex\*16, dimension( ldb, \* ), intent(inout) b, integer, intent(in) ldb, complex\*16, dimension( ldq, \* ), intent(inout) q, integer, intent(in) ldq, complex\*16, dimension( ldz, \* ), intent(inout) z, integer, intent(in) ldz, complex\*16, dimension( ldqc, \* ), intent(inout) qc, integer, intent(in) ldqc, complex\*16, dimension( ldzc, \* ), intent(inout) zc, integer, intent(in) ldzc, complex\*16, dimension( \* ), intent(inout) work, integer, intent(in) lwork, integer, intent(out) info) **ZLAQZ3** **Purpose:** ``` !> !> ZLAQZ3 Executes a single multishift QZ sweep !> ``` **Parameters** *ILSCHUR* ``` !> ILSCHUR is LOGICAL !> Determines whether or not to update the full Schur form !> ``` *ILQ* ``` !> ILQ is LOGICAL !> Determines whether or not to update the matrix Q !> ``` *ILZ* ``` !> ILZ is LOGICAL !> Determines whether or not to update the matrix Z !> ``` *N* ``` !> N is INTEGER !> The order of the matrices A, B, Q, and Z. N >= 0. !> ``` *ILO* ``` !> ILO is INTEGER !> ``` *IHI* ``` !> IHI is INTEGER !> ``` *NSHIFTS* ``` !> NSHIFTS is INTEGER !> The desired number of shifts to use !> ``` *NBLOCK\_DESIRED* ``` !> NBLOCK_DESIRED is INTEGER !> The desired size of the computational windows !> ``` *ALPHA* ``` !> ALPHA is COMPLEX*16 array. SR contains !> the alpha parts of the shifts to use. !> ``` *BETA* ``` !> BETA is COMPLEX*16 array. SS contains !> the scale of the shifts to use. !> ``` *A* ``` !> A is COMPLEX*16 array, dimension (LDA, N) !> ``` *LDA* ``` !> LDA is INTEGER !> The leading dimension of the array A. LDA >= max( 1, N ). !> ``` *B* ``` !> B is COMPLEX*16 array, dimension (LDB, N) !> ``` *LDB* ``` !> LDB is INTEGER !> The leading dimension of the array B. LDB >= max( 1, N ). !> ``` *Q* ``` !> Q is COMPLEX*16 array, dimension (LDQ, N) !> ``` *LDQ* ``` !> LDQ is INTEGER !> ``` *Z* ``` !> Z is COMPLEX*16 array, dimension (LDZ, N) !> ``` *LDZ* ``` !> LDZ is INTEGER !> ``` *QC* ``` !> QC is COMPLEX*16 array, dimension (LDQC, NBLOCK_DESIRED) !> ``` *LDQC* ``` !> LDQC is INTEGER !> ``` *ZC* ``` !> ZC is COMPLEX*16 array, dimension (LDZC, NBLOCK_DESIRED) !> ``` *LDZC* ``` !> LDZ is INTEGER !> ``` *WORK* ``` !> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) !> On exit, if INFO >= 0, WORK(1) returns the optimal LWORK. !> ``` *LWORK* ``` !> LWORK is INTEGER !> The dimension of the array WORK. LWORK >= max(1,N). !> !> If LWORK = -1, then a workspace query is assumed; the routine !> only calculates the optimal size of the WORK array, returns !> this value as the first entry of the WORK array, and no error !> message related to LWORK is issued by XERBLA. !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> ``` **Author** Thijs Steel, KU Leuven **Date** May 2020 Definition at line **204** of file **zlaqz3.f**. ## Author Generated automatically by Doxygen for LAPACK from the source code.