MQL5 リファレンス행렬과 벡터 메서드OpenBLASLeast Squares

Least Squares

The Least Squares section provides a set of versatile functions for solving overdetermined or underdetermined linear systems of the form A * X ≈ B using least squares methods. These functions support a variety of matrix types (float, double, complex, complexf) and implement several numerically stable algorithms based on standard LAPACK routines.

Supported Methods:

QR / LQ factorization (LeastSquaresSolution) — the primary approach for solving systems with full-rank matrices.
QR with compact WY representation (LeastSquaresSolutionWY) — an efficient method optimized for block operations.
SVD-based methods (for rank-deficient matrices):
- LeastSquaresSolutionDC — using a divide-and-conquer algorithm,
- LeastSquaresSolutionSVD — using classical singular value decomposition.
QR with column pivoting (QR with pivoting) (LeastSquaresSolutionQRPivot) — provides robust solutions for rank-deficient systems and computes the effective rank.
Tall-Skinny QR / Short-Wide LQ (LeastSquaresSolutionQRTallSkinny) — optimized for very tall or wide matrices.

Key Features:

Support for solving systems in various forms: with matrix A, its transpose (At), or Hermitian conjugate (AH).
Computation of residual vectors and singular values (for SVD-based methods).
Estimation of the effective rank of matrix A.
Support for both single right-hand side vectors and multiple right-hand side matrices.

Function	Action
LeastSquaresSolution	Solves overdetermined or underdetermined real / complex linear systems involving an m-by-n matrix A, or its transpose / conjugate-transpose, using a QR or LQ factorization of A. It is assumed that A has full rank. LAPACK function GELS.
LeastSquaresSolutionDC	Computes the minimum-norm solution to a real or 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. The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. LAPACK function GELSD.
LeastSquaresSolutionSVD	Computes the minimum-norm solution to a real or 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. LAPACK function GELSS.
LeastSquaresSolutionWY	Solves overdetermined or underdetermined real / complex linear systems involving an m-by-n matrix A, or its transpose / conjugate-transpose, using a QR or LQ factorization of A with compact WY representation of Q. It is assumed that A has full rank. LAPACK function GELST.
LeastSquaresSolutionsQRPivot	Computes the minimum-norm solution to a real or complex linear least squares problem: minimize 2-norm(\| b - A*x \|) using a complete orthogonal factorization of A. A is an m-by-n matrix which may be rank-deficient. LAPACK function GELSY.
LeastSquaresSolutionQRTallSkinny	Solves overdetermined or underdetermined real / complex linear systems involving an m-by-n matrix A, or its transpose / conjugate-transpose, using a tall skinny QR or short wide LQ factorization of A. It is assumed that A has full rank. LAPACK function GETSLS.
LeastSquaresGaussMarkov2	Solves a general Gauss-Markov linear model problem: minimize \|\| y \|\|_2 subject to d = Ax + By, using a generalized QR factorization of matrices A and B. LAPACK function GGGLM.

SylvesterEquationSchurBlocked

LeastSquaresSolution