- LeastSquaresSolution
- LeastSquaresSolutionDC
- LeastSquaresSolutionSVD
- LeastSquaresSolutionWY
- LeastSquaresSolutionsQRPivot
- LeastSquaresSolutionQRTallSkinny
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 |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |