- FactorizationQR
- FactorizationQRNonNeg
- FactorizationQRPivot
- FactorizationQRTallSkinny
- FactorizationLQ
- FactorizationLQShortWide
- FactorizationQL
- FactorizationRQ
- FactorizationRZ
- FactorizationQR2
- FactorizationRQ2
Orthogonal Factorizations
OpenBLAS provides a set of routines for factoring a general (m \times n) rectangular matrix (A) into the product of an orthogonal (unitary in the complex case) matrix and a triangular (or, in some cases, trapezoidal) matrix.
A real matrix (Q) is called orthogonal if (Q^TQ = I), while a complex matrix (Q) is called unitary if (Q^HQ = I). Orthogonal and unitary matrices have the important property of preserving the Euclidean norm of a vector::
||x||2 = ||Qx||2, whenever (Q) is orthogonal or unitary.
As a result, these matrices contribute to numerical stability, since they do not amplify rounding errors.
Orthogonal factorizations are widely used for solving least-squares problems. They can also be employed as preprocessing steps in the solution of eigenvalue and singular value problems.
Функция |
Выполняемое действие |
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Computes the QR factorization of a general m-by-n matrix: A = Q * R. LAPACK function GEQRF. |
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Computes the QR factorization of a general m-by-n matrix: A = Q * R. R is an upper triangular matrix with nonnegative diagonal entries. LAPACK function GEQRFP. |
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Computes the QR factorization of a general m-by-n matrix with column pivoting: A * P = Q * R. LAPACK function GEQP3. |
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Computes a blocked Tall-Skinny QR factorization of an m-by-n (m>n) matrix: A = Q * R. LAPACK function LATSQR. |
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Computes the LQ factorization of a general m-by-n matrix: A = L * Q. LAPACK function GELQF. |
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Computes a blocked Short-Wide LQ factorization of an m-by-n (m<n) matrix: A = L * Q. LAPACK function LASWLQ. |
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Computes the QL factorization of a general m-by-n matrix: A = Q * L. LAPACK function GEQLF. |
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Computes the RQ factorization of a general m-by-n matrix: A = R * Q. LAPACK function GERQF. |
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Reduces the M-by-N ( M<=N ) real or complex upper trapezoidal matrix A to upper triangular form by means of orthogonal transformations. LAPACK function TZRZF. |
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Computes the generalized QR factorization of two matrices - A of n-by-m size and B of n-by-p size: A = Q * R, B = Q * T * Z. LAPACK function GGQRF. |
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Computes the generalized RQ factorization of an m-by-n matrix A and a p-by-n matrix B: A = R * Q, B = Z * T * Q. LAPACK function GGRQF. |