-Lapack documentation +Same as function LAPACK.dgelsy, +but right hand side is a vector and not a matrix. +For details of the arguments, see documentation of dgelsy.
-- Purpose - ======= - - DGELSY computes the minimum-norm solution to a real linear least - squares problem: - minimize || A * X - B || - using a complete orthogonal factorization 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 routine first computes a QR factorization with column pivoting: - A * P = Q * [ R11 R12 ] - [ 0 R22 ] - with R11 defined as the largest leading submatrix whose estimated - condition number is less than 1/RCOND. The order of R11, RANK, - is the effective rank of A. - - Then, R22 is considered to be negligible, and R12 is annihilated - by orthogonal transformations from the right, arriving at the - complete orthogonal factorization: - A * P = Q * [ T11 0 ] * Z - [ 0 0 ] - The minimum-norm solution is then - X = P * Z' [ inv(T11)*Q1'*B ] - [ 0 ] - where Q1 consists of the first RANK columns of Q. - - This routine is basically identical to the original xGELSX except - three differences: - o The call to the subroutine xGEQPF has been substituted by - the call to the subroutine xGEQP3. This subroutine is a Blas-3 - version of the QR factorization with column pivoting. - o Matrix B (the right hand side) is updated with Blas-3. - o The permutation of matrix B (the right hand side) is faster and - more simple. - - Arguments - ========= - - M (input) INTEGER - The number of rows of the matrix A. M >= 0. - - N (input) INTEGER - The number of columns of the matrix A. N >= 0. - - NRHS (input) INTEGER - The number of right hand sides, i.e., the number of - columns of matrices B and X. NRHS >= 0. - - A (input/output) DOUBLE PRECISION array, dimension (LDA,N) - On entry, the M-by-N matrix A. - On exit, A has been overwritten by details of its - complete orthogonal factorization. - - LDA (input) INTEGER - The leading dimension of the array A. LDA >= max(1,M). - - B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) - On entry, the M-by-NRHS right hand side matrix B. - On exit, the N-by-NRHS solution matrix X. - - LDB (input) INTEGER - The leading dimension of the array B. LDB >= max(1,M,N). - - JPVT (input/output) INTEGER array, dimension (N) - On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted - to the front of AP, otherwise column i is a free column. - On exit, if JPVT(i) = k, then the i-th column of AP - was the k-th column of A. - - RCOND (input) DOUBLE PRECISION - RCOND is used to determine the effective rank of A, which - is defined as the order of the largest leading triangular - submatrix R11 in the QR factorization with pivoting of A, - whose estimated condition number < 1/RCOND. - - RANK (output) INTEGER - The effective rank of A, i.e., the order of the submatrix - R11. This is the same as the order of the submatrix T11 - in the complete orthogonal factorization of A. - - WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) - On exit, if INFO = 0, WORK(1) returns the optimal LWORK. - - LWORK (input) INTEGER - The dimension of the array WORK. - The unblocked strategy requires that: - LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ), - where MN = min( M, N ). - The block algorithm requires that: - LWORK >= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ), - where NB is an upper bound on the blocksize returned - by ILAENV for the routines DGEQP3, DTZRZF, STZRQF, DORMQR, - and DORMRZ. - - 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 (output) INTEGER - = 0: successful exit - < 0: If INFO = -i, the i-th argument had an illegal value. -")); end dgelsy_vec; @@ -7572,7 +7465,7 @@ Lapack documentation end dgetri; pure function dgetrs - "Solve a system of linear equations with the LU decomposition from dgetrf" + "Solve a system of linear equations with B matrix and the LU decomposition from dgetrf" extends Modelica.Icons.Function; input Real LU[:, size(LU, 1)] @@ -7653,7 +7546,7 @@ Lapack documentation end dgetrs; pure function dgetrs_vec - "Solve a system of linear equations with the LU decomposition from dgetrf" + "Solve a system of linear equations with b vector and the LU decomposition from dgetrf" extends Modelica.Icons.Function; input Real LU[:, size(LU, 1)] @@ -7682,55 +7575,10 @@ Lapack documentation info) annotation (Library="lapack"); annotation (Documentation(info="
-Lapack documentation +Same as function LAPACK.dgetrs, +but right hand side is a vector and not a matrix. +For details of the arguments, see documentation of dgetrs.
-- Purpose - ======= - - DGETRS solves a system of linear equations - A * X = B or A' * X = B - with a general N-by-N matrix A using the LU factorization computed - by DGETRF. - - Arguments - ========= - - TRANS (input) CHARACTER*1 - Specifies the form of the system of equations: - = 'N': A * X = B (No transpose) - = 'T': A'* X = B (Transpose) - = 'C': A'* X = B (Conjugate transpose = Transpose) - - N (input) INTEGER - The order of the matrix A. N >= 0. - - NRHS (input) INTEGER - The number of right hand sides, i.e., the number of columns - of the matrix B. NRHS >= 0. - - A (input) DOUBLE PRECISION array, dimension (LDA,N) - The factors L and U from the factorization A = P*L*U - as computed by DGETRF. - - LDA (input) INTEGER - The leading dimension of the array A. LDA >= max(1,N). - - IPIV (input) INTEGER array, dimension (N) - The pivot indices from DGETRF; for 1<=i<=N, row i of the - matrix was interchanged with row IPIV(i). - - B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) - On entry, the right hand side matrix B. - On exit, the solution matrix X. - - LDB (input) INTEGER - The leading dimension of the array B. LDB >= max(1,N). - - INFO (output) INTEGER - = 0: successful exit - < 0: if INFO = -i, the i-th argument had an illegal value -")); end dgetrs_vec;