--- name: "gglse" 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/gglse" --- # gglse(3) gglse: equality-constrained least squares ## NAME gglse - gglse: equality-constrained least squares ## SYNOPSIS ### Functions subroutine **cgglse** (m, n, p, a, lda, b, ldb, c, d, x, work, lwork, info)\ **CGGLSE solves overdetermined or underdetermined systems for OTHER matrices** subroutine **dgglse** (m, n, p, a, lda, b, ldb, c, d, x, work, lwork, info)\ **DGGLSE solves overdetermined or underdetermined systems for OTHER matrices** subroutine **sgglse** (m, n, p, a, lda, b, ldb, c, d, x, work, lwork, info)\ **SGGLSE solves overdetermined or underdetermined systems for OTHER matrices** subroutine **zgglse** (m, n, p, a, lda, b, ldb, c, d, x, work, lwork, info)\ **ZGGLSE solves overdetermined or underdetermined systems for OTHER matrices** ## Detailed Description ## Function Documentation ### subroutine cgglse (integer m, integer n, integer p, complex, dimension( lda, \* ) a, integer lda, complex, dimension( ldb, \* ) b, integer ldb, complex, dimension( \* ) c, complex, dimension( \* ) d, complex, dimension( \* ) x, complex, dimension( \* ) work, integer lwork, integer info) **CGGLSE solves overdetermined or underdetermined systems for OTHER matrices** **Purpose:** ``` !> !> CGGLSE solves the linear equality-constrained least squares (LSE) !> problem: !> !> minimize || c - A*x ||_2 subject to B*x = d !> !> where A is an M-by-N matrix, B is a P-by-N matrix, c is a given !> M-vector, and d is a given P-vector. It is assumed that !> P <= N <= M+P, and !> !> rank(B) = P and rank( (A) ) = N. !> ( (B) ) !> !> These conditions ensure that the LSE problem has a unique solution, !> which is obtained using a generalized RQ factorization of the !> matrices (B, A) given by !> !> B = (0 R)*Q, A = Z*T*Q. !> !> Callers of this subroutine should note that the singularity/rank-deficiency checks !> implemented in this subroutine are rudimentary. The CTRTRS subroutine called by this !> subroutine only signals a failure due to singularity if the problem is exactly singular. !> !> It is conceivable for one (or more) of the factors involved in the generalized RQ !> factorization of the pair (B, A) to be subnormally close to singularity without this !> subroutine signalling an error. The solutions computed for such almost-rank-deficient !> problems may be less accurate due to a loss of numerical precision. !> !> ``` **Parameters** *M* ``` !> M is INTEGER !> The number of rows of the matrix A. M >= 0. !> ``` *N* ``` !> N is INTEGER !> The number of columns of the matrices A and B. N >= 0. !> ``` *P* ``` !> P is INTEGER !> The number of rows of the matrix B. 0 <= P <= N <= M+P. !> ``` *A* ``` !> A is COMPLEX array, dimension (LDA,N) !> On entry, the M-by-N matrix A. !> On exit, the elements on and above the diagonal of the array !> contain the min(M,N)-by-N upper trapezoidal matrix T. !> ``` *LDA* ``` !> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,M). !> ``` *B* ``` !> B is COMPLEX array, dimension (LDB,N) !> On entry, the P-by-N matrix B. !> On exit, the upper triangle of the subarray B(1:P,N-P+1:N) !> contains the P-by-P upper triangular matrix R. !> ``` *LDB* ``` !> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,P). !> ``` *C* ``` !> C is COMPLEX array, dimension (M) !> On entry, C contains the right hand side vector for the !> least squares part of the LSE problem. !> On exit, the residual sum of squares for the solution !> is given by the sum of squares of elements N-P+1 to M of !> vector C. !> ``` *D* ``` !> D is COMPLEX array, dimension (P) !> On entry, D contains the right hand side vector for the !> constrained equation. !> On exit, D is destroyed. !> ``` *X* ``` !> X is COMPLEX array, dimension (N) !> On exit, X is the solution of the LSE problem. !> ``` *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,M+N+P). !> For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB, !> where NB is an upper bound for the optimal blocksizes for !> CGEQRF, CGERQF, CUNMQR and CUNMRQ. !> !> 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. !> = 1: the upper triangular factor R associated with B in the !> generalized RQ factorization of the pair (B, A) is exactly !> singular, so that rank(B) < P; the least squares !> solution could not be computed. !> = 2: the (N-P) by (N-P) part of the upper trapezoidal factor !> T associated with A in the generalized RQ factorization !> of the pair (B, A) is exactly singular, so that !> rank( (A) ) < N; the least squares solution could not !> ( (B) ) !> be computed. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **188** of file **cgglse.f**. ### subroutine dgglse (integer m, integer n, integer p, double precision, dimension( lda, \* ) a, integer lda, double precision, dimension( ldb, \* ) b, integer ldb, double precision, dimension( \* ) c, double precision, dimension( \* ) d, double precision, dimension( \* ) x, double precision, dimension( \* ) work, integer lwork, integer info) **DGGLSE solves overdetermined or underdetermined systems for OTHER matrices** **Purpose:** ``` !> !> DGGLSE solves the linear equality-constrained least squares (LSE) !> problem: !> !> minimize || c - A*x ||_2 subject to B*x = d !> !> where A is an M-by-N matrix, B is a P-by-N matrix, c is a given !> M-vector, and d is a given P-vector. It is assumed that !> P <= N <= M+P, and !> !> rank(B) = P and rank( (A) ) = N. !> ( (B) ) !> !> These conditions ensure that the LSE problem has a unique solution, !> which is obtained using a generalized RQ factorization of the !> matrices (B, A) given by !> !> B = (0 R)*Q, A = Z*T*Q. !> !> Callers of this subroutine should note that the singularity/rank-deficiency checks !> implemented in this subroutine are rudimentary. The DTRTRS subroutine called by this !> subroutine only signals a failure due to singularity if the problem is exactly singular. !> !> It is conceivable for one (or more) of the factors involved in the generalized RQ !> factorization of the pair (B, A) to be subnormally close to singularity without this !> subroutine signalling an error. The solutions computed for such almost-rank-deficient !> problems may be less accurate due to a loss of numerical precision. !> !> ``` **Parameters** *M* ``` !> M is INTEGER !> The number of rows of the matrix A. M >= 0. !> ``` *N* ``` !> N is INTEGER !> The number of columns of the matrices A and B. N >= 0. !> ``` *P* ``` !> P is INTEGER !> The number of rows of the matrix B. 0 <= P <= N <= M+P. !> ``` *A* ``` !> A is DOUBLE PRECISION array, dimension (LDA,N) !> On entry, the M-by-N matrix A. !> On exit, the elements on and above the diagonal of the array !> contain the min(M,N)-by-N upper trapezoidal matrix T. !> ``` *LDA* ``` !> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,M). !> ``` *B* ``` !> B is DOUBLE PRECISION array, dimension (LDB,N) !> On entry, the P-by-N matrix B. !> On exit, the upper triangle of the subarray B(1:P,N-P+1:N) !> contains the P-by-P upper triangular matrix R. !> ``` *LDB* ``` !> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,P). !> ``` *C* ``` !> C is DOUBLE PRECISION array, dimension (M) !> On entry, C contains the right hand side vector for the !> least squares part of the LSE problem. !> On exit, the residual sum of squares for the solution !> is given by the sum of squares of elements N-P+1 to M of !> vector C. !> ``` *D* ``` !> D is DOUBLE PRECISION array, dimension (P) !> On entry, D contains the right hand side vector for the !> constrained equation. !> On exit, D is destroyed. !> ``` *X* ``` !> X is DOUBLE PRECISION array, dimension (N) !> On exit, X is the solution of the LSE problem. !> ``` *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,M+N+P). !> For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB, !> where NB is an upper bound for the optimal blocksizes for !> DGEQRF, SGERQF, DORMQR and SORMRQ. !> !> 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. !> = 1: the upper triangular factor R associated with B in the !> generalized RQ factorization of the pair (B, A) is exactly !> singular, so that rank(B) < P; the least squares !> solution could not be computed. !> = 2: the (N-P) by (N-P) part of the upper trapezoidal factor !> T associated with A in the generalized RQ factorization !> of the pair (B, A) is exactly singular, so that !> rank( (A) ) < N; the least squares solution could not !> ( (B) ) !> be computed. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **188** of file **dgglse.f**. ### subroutine sgglse (integer m, integer n, integer p, real, dimension( lda, \* ) a, integer lda, real, dimension( ldb, \* ) b, integer ldb, real, dimension( \* ) c, real, dimension( \* ) d, real, dimension( \* ) x, real, dimension( \* ) work, integer lwork, integer info) **SGGLSE solves overdetermined or underdetermined systems for OTHER matrices** **Purpose:** ``` !> !> SGGLSE solves the linear equality-constrained least squares (LSE) !> problem: !> !> minimize || c - A*x ||_2 subject to B*x = d !> !> where A is an M-by-N matrix, B is a P-by-N matrix, c is a given !> M-vector, and d is a given P-vector. It is assumed that !> P <= N <= M+P, and !> !> rank(B) = P and rank( (A) ) = N. !> ( (B) ) !> !> These conditions ensure that the LSE problem has a unique solution, !> which is obtained using a generalized RQ factorization of the !> matrices (B, A) given by !> !> B = (0 R)*Q, A = Z*T*Q. !> !> Callers of this subroutine should note that the singularity/rank-deficiency checks !> implemented in this subroutine are rudimentary. The STRTRS subroutine called by this !> subroutine only signals a failure due to singularity if the problem is exactly singular. !> !> It is conceivable for one (or more) of the factors involved in the generalized RQ !> factorization of the pair (B, A) to be subnormally close to singularity without this !> subroutine signalling an error. The solutions computed for such almost-rank-deficient !> problems may be less accurate due to a loss of numerical precision. !> !> ``` **Parameters** *M* ``` !> M is INTEGER !> The number of rows of the matrix A. M >= 0. !> ``` *N* ``` !> N is INTEGER !> The number of columns of the matrices A and B. N >= 0. !> ``` *P* ``` !> P is INTEGER !> The number of rows of the matrix B. 0 <= P <= N <= M+P. !> ``` *A* ``` !> A is REAL array, dimension (LDA,N) !> On entry, the M-by-N matrix A. !> On exit, the elements on and above the diagonal of the array !> contain the min(M,N)-by-N upper trapezoidal matrix T. !> ``` *LDA* ``` !> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,M). !> ``` *B* ``` !> B is REAL array, dimension (LDB,N) !> On entry, the P-by-N matrix B. !> On exit, the upper triangle of the subarray B(1:P,N-P+1:N) !> contains the P-by-P upper triangular matrix R. !> ``` *LDB* ``` !> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,P). !> ``` *C* ``` !> C is REAL array, dimension (M) !> On entry, C contains the right hand side vector for the !> least squares part of the LSE problem. !> On exit, the residual sum of squares for the solution !> is given by the sum of squares of elements N-P+1 to M of !> vector C. !> ``` *D* ``` !> D is REAL array, dimension (P) !> On entry, D contains the right hand side vector for the !> constrained equation. !> On exit, D is destroyed. !> ``` *X* ``` !> X is REAL array, dimension (N) !> On exit, X is the solution of the LSE problem. !> ``` *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,M+N+P). !> For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB, !> where NB is an upper bound for the optimal blocksizes for !> SGEQRF, SGERQF, SORMQR and SORMRQ. !> !> 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. !> = 1: the upper triangular factor R associated with B in the !> generalized RQ factorization of the pair (B, A) is exactly !> singular, so that rank(B) < P; the least squares !> solution could not be computed. !> = 2: the (N-P) by (N-P) part of the upper trapezoidal factor !> T associated with A in the generalized RQ factorization !> of the pair (B, A) is exactly singular, so that !> rank( (A) ) < N; the least squares solution could not !> ( (B) ) !> be computed. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **188** of file **sgglse.f**. ### subroutine zgglse (integer m, integer n, integer p, complex\*16, dimension( lda, \* ) a, integer lda, complex\*16, dimension( ldb, \* ) b, integer ldb, complex\*16, dimension( \* ) c, complex\*16, dimension( \* ) d, complex\*16, dimension( \* ) x, complex\*16, dimension( \* ) work, integer lwork, integer info) **ZGGLSE solves overdetermined or underdetermined systems for OTHER matrices** **Purpose:** ``` !> !> ZGGLSE solves the linear equality-constrained least squares (LSE) !> problem: !> !> minimize || c - A*x ||_2 subject to B*x = d !> !> where A is an M-by-N matrix, B is a P-by-N matrix, c is a given !> M-vector, and d is a given P-vector. It is assumed that !> P <= N <= M+P, and !> !> rank(B) = P and rank( (A) ) = N. !> ( (B) ) !> !> These conditions ensure that the LSE problem has a unique solution, !> which is obtained using a generalized RQ factorization of the !> matrices (B, A) given by !> !> B = (0 R)*Q, A = Z*T*Q. !> !> Callers of this subroutine should note that the singularity/rank-deficiency checks !> implemented in this subroutine are rudimentary. The ZTRTRS subroutine called by this !> subroutine only signals a failure due to singularity if the problem is exactly singular. !> !> It is conceivable for one (or more) of the factors involved in the generalized RQ !> factorization of the pair (B, A) to be subnormally close to singularity without this !> subroutine signalling an error. The solutions computed for such almost-rank-deficient !> problems may be less accurate due to a loss of numerical precision. !> !> ``` **Parameters** *M* ``` !> M is INTEGER !> The number of rows of the matrix A. M >= 0. !> ``` *N* ``` !> N is INTEGER !> The number of columns of the matrices A and B. N >= 0. !> ``` *P* ``` !> P is INTEGER !> The number of rows of the matrix B. 0 <= P <= N <= M+P. !> ``` *A* ``` !> A is COMPLEX*16 array, dimension (LDA,N) !> On entry, the M-by-N matrix A. !> On exit, the elements on and above the diagonal of the array !> contain the min(M,N)-by-N upper trapezoidal matrix T. !> ``` *LDA* ``` !> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,M). !> ``` *B* ``` !> B is COMPLEX*16 array, dimension (LDB,N) !> On entry, the P-by-N matrix B. !> On exit, the upper triangle of the subarray B(1:P,N-P+1:N) !> contains the P-by-P upper triangular matrix R. !> ``` *LDB* ``` !> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,P). !> ``` *C* ``` !> C is COMPLEX*16 array, dimension (M) !> On entry, C contains the right hand side vector for the !> least squares part of the LSE problem. !> On exit, the residual sum of squares for the solution !> is given by the sum of squares of elements N-P+1 to M of !> vector C. !> ``` *D* ``` !> D is COMPLEX*16 array, dimension (P) !> On entry, D contains the right hand side vector for the !> constrained equation. !> On exit, D is destroyed. !> ``` *X* ``` !> X is COMPLEX*16 array, dimension (N) !> On exit, X is the solution of the LSE problem. !> ``` *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,M+N+P). !> For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB, !> where NB is an upper bound for the optimal blocksizes for !> ZGEQRF, CGERQF, ZUNMQR and CUNMRQ. !> !> 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. !> = 1: the upper triangular factor R associated with B in the !> generalized RQ factorization of the pair (B, A) is exactly !> singular, so that rank(B) < P; the least squares !> solution could not be computed. !> = 2: the (N-P) by (N-P) part of the upper trapezoidal factor !> T associated with A in the generalized RQ factorization !> of the pair (B, A) is exactly singular, so that !> rank( (A) ) < N; the least squares solution could not !> ( (B) ) !> be computed. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **188** of file **zgglse.f**. ## Author Generated automatically by Doxygen for LAPACK from the source code.