--- name: "laqz2" 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/laqz2" --- # laqz2(3) laqz2: step in ggev3, gges3 ## NAME laqz2 - laqz2: step in ggev3, gges3 ## SYNOPSIS ### Functions recursive subroutine **claqz2** (ilschur, ilq, ilz, n, ilo, ihi, nw, a, lda, b, ldb, q, ldq, z, ldz, ns, nd, alpha, beta, qc, ldqc, zc, ldzc, work, lwork, rwork, rec, info)\ **CLAQZ2** subroutine **dlaqz2** (ilq, ilz, k, istartm, istopm, ihi, a, lda, b, ldb, nq, qstart, q, ldq, nz, zstart, z, ldz)\ **DLAQZ2** subroutine **slaqz2** (ilq, ilz, k, istartm, istopm, ihi, a, lda, b, ldb, nq, qstart, q, ldq, nz, zstart, z, ldz)\ **SLAQZ2** recursive subroutine **zlaqz2** (ilschur, ilq, ilz, n, ilo, ihi, nw, a, lda, b, ldb, q, ldq, z, ldz, ns, nd, alpha, beta, qc, ldqc, zc, ldzc, work, lwork, rwork, rec, info)\ **ZLAQZ2** ## Detailed Description ## Function Documentation ### recursive subroutine claqz2 (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, 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, integer, intent(out) ns, integer, intent(out) nd, complex, dimension( \* ), intent(inout) alpha, complex, dimension( \* ), intent(inout) beta, complex, dimension( ldqc, \* ) qc, integer, intent(in) ldqc, complex, dimension( ldzc, \* ) zc, integer, intent(in) ldzc, complex, dimension( \* ) work, integer, intent(in) lwork, real, dimension( \* ) rwork, integer, intent(in) rec, integer, intent(out) info) **CLAQZ2** **Purpose:** ``` !> !> CLAQZ2 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 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 !> ``` *NS* ``` !> NS is INTEGER !> The number of unconverged eigenvalues available to !> use as shifts. !> ``` *ND* ``` !> ND is INTEGER !> The number of converged eigenvalues found. !> ``` *ALPHA* ``` !> ALPHA is COMPLEX array, dimension (N) !> Each scalar alpha defining an eigenvalue !> of GNEP. !> ``` *BETA* ``` !> BETA is COMPLEX array, dimension (N) !> The scalars beta that define the eigenvalues of GNEP. !> Together, the quantities alpha = ALPHA(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 COMPLEX array, dimension (LDQC, NW) !> ``` *LDQC* ``` !> LDQC is INTEGER !> ``` *ZC* ``` !> ZC is COMPLEX array, dimension (LDZC, NW) !> ``` *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. !> ``` *RWORK* ``` !> RWORK is REAL array, dimension (N) !> ``` *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, KU Leuven **Date** May 2020 Definition at line **231** of file **claqz2.f**. ### subroutine dlaqz2 (logical, intent(in) ilq, logical, intent(in) ilz, integer, intent(in) k, integer, intent(in) istartm, integer, intent(in) istopm, integer, intent(in) ihi, double precision, dimension( lda, \* ) a, integer, intent(in) lda, double precision, dimension( ldb, \* ) b, integer, intent(in) ldb, integer, intent(in) nq, integer, intent(in) qstart, double precision, dimension( ldq, \* ) q, integer, intent(in) ldq, integer, intent(in) nz, integer, intent(in) zstart, double precision, dimension( ldz, \* ) z, integer, intent(in) ldz) **DLAQZ2** **Purpose:** ``` !> !> DLAQZ2 chases a 2x2 shift bulge in a matrix pencil down a single position !> ``` **Parameters** *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 !> ``` *K* ``` !> K is INTEGER !> Index indicating the position of the bulge. !> On entry, the bulge is located in !> (A(k+1:k+2,k:k+1),B(k+1:k+2,k:k+1)). !> On exit, the bulge is located in !> (A(k+2:k+3,k+1:k+2),B(k+2:k+3,k+1:k+2)). !> ``` *ISTARTM* ``` !> ISTARTM is INTEGER !> ``` *ISTOPM* ``` !> ISTOPM is INTEGER !> Updates to (A,B) are restricted to !> (istartm:k+3,k:istopm). It is assumed !> without checking that istartm <= k+1 and !> k+2 <= istopm !> ``` *IHI* ``` !> IHI is INTEGER !> ``` *A* ``` !> A is DOUBLE PRECISION array, dimension (LDA,N) !> ``` *LDA* ``` !> LDA is INTEGER !> The leading dimension of A as declared in !> the calling procedure. !> ``` *B* ``` !> B is DOUBLE PRECISION array, dimension (LDB,N) !> ``` *LDB* ``` !> LDB is INTEGER !> The leading dimension of B as declared in !> the calling procedure. !> ``` *NQ* ``` !> NQ is INTEGER !> The order of the matrix Q !> ``` *QSTART* ``` !> QSTART is INTEGER !> Start index of the matrix Q. Rotations are applied !> To columns k+2-qStart:k+4-qStart of Q. !> ``` *Q* ``` !> Q is DOUBLE PRECISION array, dimension (LDQ,NQ) !> ``` *LDQ* ``` !> LDQ is INTEGER !> The leading dimension of Q as declared in !> the calling procedure. !> ``` *NZ* ``` !> NZ is INTEGER !> The order of the matrix Z !> ``` *ZSTART* ``` !> ZSTART is INTEGER !> Start index of the matrix Z. Rotations are applied !> To columns k+1-qStart:k+3-qStart of Z. !> ``` *Z* ``` !> Z is DOUBLE PRECISION array, dimension (LDZ,NZ) !> ``` *LDZ* ``` !> LDZ is INTEGER !> The leading dimension of Q as declared in !> the calling procedure. !> ``` **Author** Thijs Steel, KU Leuven **Date** May 2020 Definition at line **172** of file **dlaqz2.f**. ### subroutine slaqz2 (logical, intent(in) ilq, logical, intent(in) ilz, integer, intent(in) k, integer, intent(in) istartm, integer, intent(in) istopm, integer, intent(in) ihi, real, dimension( lda, \* ) a, integer, intent(in) lda, real, dimension( ldb, \* ) b, integer, intent(in) ldb, integer, intent(in) nq, integer, intent(in) qstart, real, dimension( ldq, \* ) q, integer, intent(in) ldq, integer, intent(in) nz, integer, intent(in) zstart, real, dimension( ldz, \* ) z, integer, intent(in) ldz) **SLAQZ2** **Purpose:** ``` !> !> SLAQZ2 chases a 2x2 shift bulge in a matrix pencil down a single position !> ``` **Parameters** *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 !> ``` *K* ``` !> K is INTEGER !> Index indicating the position of the bulge. !> On entry, the bulge is located in !> (A(k+1:k+2,k:k+1),B(k+1:k+2,k:k+1)). !> On exit, the bulge is located in !> (A(k+2:k+3,k+1:k+2),B(k+2:k+3,k+1:k+2)). !> ``` *ISTARTM* ``` !> ISTARTM is INTEGER !> ``` *ISTOPM* ``` !> ISTOPM is INTEGER !> Updates to (A,B) are restricted to !> (istartm:k+3,k:istopm). It is assumed !> without checking that istartm <= k+1 and !> k+2 <= istopm !> ``` *IHI* ``` !> IHI is INTEGER !> ``` *A* ``` !> A is REAL array, dimension (LDA,N) !> ``` *LDA* ``` !> LDA is INTEGER !> The leading dimension of A as declared in !> the calling procedure. !> ``` *B* ``` !> B is REAL array, dimension (LDB,N) !> ``` *LDB* ``` !> LDB is INTEGER !> The leading dimension of B as declared in !> the calling procedure. !> ``` *NQ* ``` !> NQ is INTEGER !> The order of the matrix Q !> ``` *QSTART* ``` !> QSTART is INTEGER !> Start index of the matrix Q. Rotations are applied !> To columns k+2-qStart:k+4-qStart of Q. !> ``` *Q* ``` !> Q is REAL array, dimension (LDQ,NQ) !> ``` *LDQ* ``` !> LDQ is INTEGER !> The leading dimension of Q as declared in !> the calling procedure. !> ``` *NZ* ``` !> NZ is INTEGER !> The order of the matrix Z !> ``` *ZSTART* ``` !> ZSTART is INTEGER !> Start index of the matrix Z. Rotations are applied !> To columns k+1-qStart:k+3-qStart of Z. !> ``` *Z* ``` !> Z is REAL array, dimension (LDZ,NZ) !> ``` *LDZ* ``` !> LDZ is INTEGER !> The leading dimension of Q as declared in !> the calling procedure. !> ``` **Author** Thijs Steel, KU Leuven **Date** May 2020 Definition at line **171** of file **slaqz2.f**. ### recursive subroutine zlaqz2 (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, 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, integer, intent(out) ns, integer, intent(out) nd, complex\*16, dimension( \* ), intent(inout) alpha, complex\*16, dimension( \* ), intent(inout) beta, complex\*16, dimension( ldqc, \* ) qc, integer, intent(in) ldqc, complex\*16, dimension( ldzc, \* ) zc, integer, intent(in) ldzc, complex\*16, dimension( \* ) work, integer, intent(in) lwork, double precision, dimension( \* ) rwork, integer, intent(in) rec, integer, intent(out) info) **ZLAQZ2** **Purpose:** ``` !> !> ZLAQZ2 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 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 !> ``` *NS* ``` !> NS is INTEGER !> The number of unconverged eigenvalues available to !> use as shifts. !> ``` *ND* ``` !> ND is INTEGER !> The number of converged eigenvalues found. !> ``` *ALPHA* ``` !> ALPHA is COMPLEX*16 array, dimension (N) !> Each scalar alpha defining an eigenvalue !> of GNEP. !> ``` *BETA* ``` !> BETA is COMPLEX*16 array, dimension (N) !> The scalars beta that define the eigenvalues of GNEP. !> Together, the quantities alpha = ALPHA(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 COMPLEX*16 array, dimension (LDQC, NW) !> ``` *LDQC* ``` !> LDQC is INTEGER !> ``` *ZC* ``` !> ZC is COMPLEX*16 array, dimension (LDZC, NW) !> ``` *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. !> ``` *RWORK* ``` !> RWORK is DOUBLE PRECISION array, dimension (N) !> ``` *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 **230** of file **zlaqz2.f**. ## Author Generated automatically by Doxygen for LAPACK from the source code.