--- name: "gelss" 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/gelss" --- # gelss(3) gelss: least squares using SVD, QR iteration ## NAME gelss - gelss: least squares using SVD, QR iteration ## SYNOPSIS ### Functions subroutine **cgelss** (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, rwork, info)\ **CGELSS solves overdetermined or underdetermined systems for GE matrices** subroutine **dgelss** (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, info)\ **DGELSS solves overdetermined or underdetermined systems for GE matrices** subroutine **sgelss** (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, info)\ **SGELSS solves overdetermined or underdetermined systems for GE matrices** subroutine **zgelss** (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, rwork, info)\ **ZGELSS solves overdetermined or underdetermined systems for GE matrices** ## Detailed Description ## Function Documentation ### subroutine cgelss (integer m, integer n, integer nrhs, complex, dimension( lda, \* ) a, integer lda, complex, dimension( ldb, \* ) b, integer ldb, real, dimension( \* ) s, real rcond, integer rank, complex, dimension( \* ) work, integer lwork, real, dimension( \* ) rwork, integer info) **CGELSS solves overdetermined or underdetermined systems for GE matrices** **Purpose:** ``` !> !> CGELSS computes the minimum norm solution to a complex linear !> least squares problem: !> !> Minimize 2-norm(| b - A*x |). !> !> using the singular value decomposition (SVD) of A. A is an M-by-N !> matrix which may be rank-deficient. !> !> Several right hand side vectors b and solution vectors x can be !> handled in a single call; they are stored as the columns of the !> M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix !> X. !> !> The effective rank of A is determined by treating as zero those !> singular values which are less than RCOND times the largest singular !> value. !> ``` **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 matrix A. N >= 0. !> ``` *NRHS* ``` !> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrices B and X. NRHS >= 0. !> ``` *A* ``` !> A is COMPLEX array, dimension (LDA,N) !> On entry, the M-by-N matrix A. !> On exit, the first min(m,n) rows of A are overwritten with !> its right singular vectors, stored rowwise. !> ``` *LDA* ``` !> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,M). !> ``` *B* ``` !> B is COMPLEX array, dimension (LDB,NRHS) !> On entry, the M-by-NRHS right hand side matrix B. !> On exit, B is overwritten by the N-by-NRHS solution matrix X. !> If m >= n and RANK = n, the residual sum-of-squares for !> the solution in the i-th column is given by the sum of !> squares of the modulus of elements n+1:m in that column. !> ``` *LDB* ``` !> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,M,N). !> ``` *S* ``` !> S is REAL array, dimension (min(M,N)) !> The singular values of A in decreasing order. !> The condition number of A in the 2-norm = S(1)/S(min(m,n)). !> ``` *RCOND* ``` !> RCOND is REAL !> RCOND is used to determine the effective rank of A. !> Singular values S(i) <= RCOND*S(1) are treated as zero. !> If RCOND < 0, machine precision is used instead. !> ``` *RANK* ``` !> RANK is INTEGER !> The effective rank of A, i.e., the number of singular values !> which are greater than RCOND*S(1). !> ``` *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 >= 1, and also: !> LWORK >= 2*min(M,N) + max(M,N,NRHS) !> For good performance, LWORK should generally be larger. !> !> 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 (5*min(M,N)) !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value. !> > 0: the algorithm for computing the SVD failed to converge; !> if INFO = i, i off-diagonal elements of an intermediate !> bidiagonal form did not converge to zero. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **176** of file **cgelss.f**. ### subroutine dgelss (integer m, integer n, integer nrhs, double precision, dimension( lda, \* ) a, integer lda, double precision, dimension( ldb, \* ) b, integer ldb, double precision, dimension( \* ) s, double precision rcond, integer rank, double precision, dimension( \* ) work, integer lwork, integer info) **DGELSS solves overdetermined or underdetermined systems for GE matrices** **Purpose:** ``` !> !> DGELSS computes the minimum norm solution to a real linear least !> squares problem: !> !> Minimize 2-norm(| b - A*x |). !> !> using the singular value decomposition (SVD) of A. A is an M-by-N !> matrix which may be rank-deficient. !> !> Several right hand side vectors b and solution vectors x can be !> handled in a single call; they are stored as the columns of the !> M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix !> X. !> !> The effective rank of A is determined by treating as zero those !> singular values which are less than RCOND times the largest singular !> value. !> ``` **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 matrix A. N >= 0. !> ``` *NRHS* ``` !> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrices B and X. NRHS >= 0. !> ``` *A* ``` !> A is DOUBLE PRECISION array, dimension (LDA,N) !> On entry, the M-by-N matrix A. !> On exit, the first min(m,n) rows of A are overwritten with !> its right singular vectors, stored rowwise. !> ``` *LDA* ``` !> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,M). !> ``` *B* ``` !> B is DOUBLE PRECISION array, dimension (LDB,NRHS) !> On entry, the M-by-NRHS right hand side matrix B. !> On exit, B is overwritten by the N-by-NRHS solution !> matrix X. If m >= n and RANK = n, the residual !> sum-of-squares for the solution in the i-th column is given !> by the sum of squares of elements n+1:m in that column. !> ``` *LDB* ``` !> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,max(M,N)). !> ``` *S* ``` !> S is DOUBLE PRECISION array, dimension (min(M,N)) !> The singular values of A in decreasing order. !> The condition number of A in the 2-norm = S(1)/S(min(m,n)). !> ``` *RCOND* ``` !> RCOND is DOUBLE PRECISION !> RCOND is used to determine the effective rank of A. !> Singular values S(i) <= RCOND*S(1) are treated as zero. !> If RCOND < 0, machine precision is used instead. !> ``` *RANK* ``` !> RANK is INTEGER !> The effective rank of A, i.e., the number of singular values !> which are greater than RCOND*S(1). !> ``` *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 >= 1, and also: !> LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS ) !> For good performance, LWORK should generally be larger. !> !> 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. !> > 0: the algorithm for computing the SVD failed to converge; !> if INFO = i, i off-diagonal elements of an intermediate !> bidiagonal form did not converge to zero. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **170** of file **dgelss.f**. ### subroutine sgelss (integer m, integer n, integer nrhs, real, dimension( lda, \* ) a, integer lda, real, dimension( ldb, \* ) b, integer ldb, real, dimension( \* ) s, real rcond, integer rank, real, dimension( \* ) work, integer lwork, integer info) **SGELSS solves overdetermined or underdetermined systems for GE matrices** **Purpose:** ``` !> !> SGELSS computes the minimum norm solution to a real linear least !> squares problem: !> !> Minimize 2-norm(| b - A*x |). !> !> using the singular value decomposition (SVD) of A. A is an M-by-N !> matrix which may be rank-deficient. !> !> Several right hand side vectors b and solution vectors x can be !> handled in a single call; they are stored as the columns of the !> M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix !> X. !> !> The effective rank of A is determined by treating as zero those !> singular values which are less than RCOND times the largest singular !> value. !> ``` **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 matrix A. N >= 0. !> ``` *NRHS* ``` !> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrices B and X. NRHS >= 0. !> ``` *A* ``` !> A is REAL array, dimension (LDA,N) !> On entry, the M-by-N matrix A. !> On exit, the first min(m,n) rows of A are overwritten with !> its right singular vectors, stored rowwise. !> ``` *LDA* ``` !> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,M). !> ``` *B* ``` !> B is REAL array, dimension (LDB,NRHS) !> On entry, the M-by-NRHS right hand side matrix B. !> On exit, B is overwritten by the N-by-NRHS solution !> matrix X. If m >= n and RANK = n, the residual !> sum-of-squares for the solution in the i-th column is given !> by the sum of squares of elements n+1:m in that column. !> ``` *LDB* ``` !> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,max(M,N)). !> ``` *S* ``` !> S is REAL array, dimension (min(M,N)) !> The singular values of A in decreasing order. !> The condition number of A in the 2-norm = S(1)/S(min(m,n)). !> ``` *RCOND* ``` !> RCOND is REAL !> RCOND is used to determine the effective rank of A. !> Singular values S(i) <= RCOND*S(1) are treated as zero. !> If RCOND < 0, machine precision is used instead. !> ``` *RANK* ``` !> RANK is INTEGER !> The effective rank of A, i.e., the number of singular values !> which are greater than RCOND*S(1). !> ``` *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 >= 1, and also: !> LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS ) !> For good performance, LWORK should generally be larger. !> !> 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. !> > 0: the algorithm for computing the SVD failed to converge; !> if INFO = i, i off-diagonal elements of an intermediate !> bidiagonal form did not converge to zero. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **170** of file **sgelss.f**. ### subroutine zgelss (integer m, integer n, integer nrhs, complex\*16, dimension( lda, \* ) a, integer lda, complex\*16, dimension( ldb, \* ) b, integer ldb, double precision, dimension( \* ) s, double precision rcond, integer rank, complex\*16, dimension( \* ) work, integer lwork, double precision, dimension( \* ) rwork, integer info) **ZGELSS solves overdetermined or underdetermined systems for GE matrices** **Purpose:** ``` !> !> ZGELSS computes the minimum norm solution to a complex linear !> least squares problem: !> !> Minimize 2-norm(| b - A*x |). !> !> using the singular value decomposition (SVD) of A. A is an M-by-N !> matrix which may be rank-deficient. !> !> Several right hand side vectors b and solution vectors x can be !> handled in a single call; they are stored as the columns of the !> M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix !> X. !> !> The effective rank of A is determined by treating as zero those !> singular values which are less than RCOND times the largest singular !> value. !> ``` **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 matrix A. N >= 0. !> ``` *NRHS* ``` !> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrices B and X. NRHS >= 0. !> ``` *A* ``` !> A is COMPLEX*16 array, dimension (LDA,N) !> On entry, the M-by-N matrix A. !> On exit, the first min(m,n) rows of A are overwritten with !> its right singular vectors, stored rowwise. !> ``` *LDA* ``` !> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,M). !> ``` *B* ``` !> B is COMPLEX*16 array, dimension (LDB,NRHS) !> On entry, the M-by-NRHS right hand side matrix B. !> On exit, B is overwritten by the N-by-NRHS solution matrix X. !> If m >= n and RANK = n, the residual sum-of-squares for !> the solution in the i-th column is given by the sum of !> squares of the modulus of elements n+1:m in that column. !> ``` *LDB* ``` !> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,M,N). !> ``` *S* ``` !> S is DOUBLE PRECISION array, dimension (min(M,N)) !> The singular values of A in decreasing order. !> The condition number of A in the 2-norm = S(1)/S(min(m,n)). !> ``` *RCOND* ``` !> RCOND is DOUBLE PRECISION !> RCOND is used to determine the effective rank of A. !> Singular values S(i) <= RCOND*S(1) are treated as zero. !> If RCOND < 0, machine precision is used instead. !> ``` *RANK* ``` !> RANK is INTEGER !> The effective rank of A, i.e., the number of singular values !> which are greater than RCOND*S(1). !> ``` *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 >= 1, and also: !> LWORK >= 2*min(M,N) + max(M,N,NRHS) !> For good performance, LWORK should generally be larger. !> !> 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 (5*min(M,N)) !> ``` *INFO* ``` !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value. !> > 0: the algorithm for computing the SVD failed to converge; !> if INFO = i, i off-diagonal elements of an intermediate !> bidiagonal form did not converge to zero. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **176** of file **zgelss.f**. ## Author Generated automatically by Doxygen for LAPACK from the source code.