linalg_qr Module

On input, an M-by-N matrix where the elements below the diagonal contain the elementary reflectors generated from the QR factorization. On and above the diagonal, the matrix contains the matrix $R$. On output, the elements below the diagonal are zeroed such that the remaining matrix is simply the M-by-N matrix $R$.

A MIN(M, N)-element array containing the scalar factors of each elementary reflector defined in $R$.

An M-by-M matrix where the full orthogonal matrix $Q$ will be written. In the event that M > N, $Q$ may be supplied as M-by-N, and therefore only return the useful submatrix $Q_1$ $Q = [Q_1 Q_2]$ as the factorization can be written as $Q R = [Q_1, Q_2] [R1 0]^T$.

An optional input, that if provided, prevents any local memory allocation. If not provided, the memory required is allocated within. If provided, the length of the array must be at least olwork.

An optional output used to determine workspace size. If supplied, the routine determines the optimal size for work, and returns without performing any actual calculations.

The error object to be updated.

On input, an M-by-N matrix where the elements below the diagonal contain the elementary reflectors generated from the QR factorization. On and above the diagonal, the matrix contains the matrix $R$. On output, the elements below the diagonal are zeroed such that the remaining matrix is simply the M-by-N matrix $R$.

A MIN(M, N)-element array containing the scalar factors of each elementary reflector defined in $R$.

An M-by-M matrix where the full orthogonal matrix $Q$ will be written. In the event that M > N, $Q$ may be supplied as M-by-N, and therefore only return the useful submatrix $Q_1$ $Q = [Q_1 Q_2]$ as the factorization can be written as $Q R = [Q_1, Q_2] [R1 0]^T$.

An optional input, that if provided, prevents any local memory allocation. If not provided, the memory required is allocated within. If provided, the length of the array must be at least olwork.

An optional output used to determine workspace size. If supplied, the routine determines the optimal size for work, and returns without performing any actual calculations.

The error object to be updated.

On input, an M-by-N matrix where the elements below the diagonal contain the elementary reflectors generated from the QR factorization. On and above the diagonal, the matrix contains the matrix $R$. On output, the elements below the diagonal are zeroed such that the remaining matrix is simply the M-by-N matrix $R$.

A MIN(M, N)-element array containing the scalar factors of each elementary reflector defined in $R$.

An N-element column pivot array as returned by the QR factorization.

An M-by-M matrix where the full orthogonal matrix $Q$ will be written. In the event that M > N, $Q$ may be supplied as M-by-N, and therefore only return the useful submatrix $Q_1$ $Q = [Q_1 Q_2]$ as the factorization can be written as $Q R = [Q_1, Q_2] [R1 0]^T$.

An N-by-N matrix where the pivot matrix will be written.

An optional input, that if provided, prevents any local memory allocation. If not provided, the memory required is allocated within. If provided, the length of the array must be at least olwork.

An optional output used to determine workspace size. If supplied, the routine determines the optimal size for work, and returns without performing any actual calculations.

The error object to be updated.

On input, an M-by-N matrix where the elements below the diagonal contain the elementary reflectors generated from the QR factorization. On and above the diagonal, the matrix contains the matrix $R$. On output, the elements below the diagonal are zeroed such that the remaining matrix is simply the M-by-N matrix $R$.

A MIN(M, N)-element array containing the scalar factors of each elementary reflector defined in $R$.

An N-element column pivot array as returned by the QR factorization.

An M-by-M matrix where the full orthogonal matrix $Q$ will be written. In the event that M > N, $Q$ may be supplied as M-by-N, and therefore only return the useful submatrix $Q_1$ $Q = [Q_1 Q_2]$ as the factorization can be written as $Q R = [Q_1, Q_2] [R1 0]^T$.

An N-by-N matrix where the pivot matrix will be written.

An optional input, that if provided, prevents any local memory allocation. If not provided, the memory required is allocated within. If provided, the length of the array must be at least olwork.

An optional output used to determine workspace size. If supplied, the routine determines the optimal size for work, and returns without performing any actual calculations.

The error object to be updated.

Set to true to apply $Q$ or $Q^T$ from the left; else, set to false to apply $Q$ or $Q^T$ from the right.

Set to true to apply $Q^T$; else, set to false to apply $Q$.

On input, an LDA-by-K matrix containing the elementary reflectors output from the QR factorization. If lside is set to true, LDA = M, and M >= K >= 0; else, if lside is set to false, LDA = N, and N >= K >= 0. Notice, the contents of this matrix are restored on exit.

A K-element array containing the scalar factors of each elementary reflector defined in$A$.

On input, the M-by-N matrix $C$. On output, the product of the orthogonal matrix $Q$ and the original matrix $C$.

An optional input, that if provided, prevents any local memory allocation. If not provided, the memory required is allocated within. If provided, the length of the array must be at least olwork.

An optional output used to determine workspace size. If supplied, the routine determines the optimal size for work, and returns without performing any actual calculations.

The error object to be updated.

Set to true to apply $Q$ or $Q^H$ from the left; else, set to false to apply $Q$ or $Q^H$ from the right.

Set to true to apply $Q^H$; else, set to false to apply $Q$.

On input, an LDA-by-K matrix containing the elementary reflectors output from the QR factorization. If lside is set to true, LDA = M, and M >= K >= 0; else, if lside is set to false, LDA = N, and N >= K >= 0. Notice, the contents of this matrix are restored on exit.

A K-element array containing the scalar factors of each elementary reflector defined in$A$.

On input, the M-by-N matrix $C$. On output, the product of the orthogonal matrix $Q$ and the original matrix $C$.

An optional input, that if provided, prevents any local memory allocation. If not provided, the memory required is allocated within. If provided, the length of the array must be at least olwork.

An optional output used to determine workspace size. If supplied, the routine determines the optimal size for work, and returns without performing any actual calculations.

The error object to be updated.

Set to true to apply $Q^T$; else, set to false to apply $Q$.

On input, an M-by-K matrix containing the elementary reflectors output from the QR factorization. Notice, the contents of this matrix are restored on exit.

A K-element array containing the scalar factors of each elementary reflector defined in$A$.

On input, the M-element vector $\vec{c}$. On output, the product of the orthogonal matrix $Q$ and the original vector $\vec{c}$.

An optional input, that if provided, prevents any local memory allocation. If not provided, the memory required is allocated within. If provided, the length of the array must be at least olwork.

An optional output used to determine workspace size. If supplied, the routine determines the optimal size for work, and returns without performing any actual calculations.

The error object to be updated.

Set to true to apply $Q^H$; else, set to false to apply $Q$.

On input, an M-by-K matrix containing the elementary reflectors output from the QR factorization. Notice, the contents of this matrix are restored on exit.

A K-element array containing the scalar factors of each elementary reflector defined in$A$.

On input, the M-element vector $\vec{c}$. On output, the product of the orthogonal matrix $Q$ and the original vector $\vec{c}$.

An optional input, that if provided, prevents any local memory allocation. If not provided, the memory required is allocated within. If provided, the length of the array must be at least olwork.

An optional output used to determine workspace size. If supplied, the routine determines the optimal size for work, and returns without performing any actual calculations.

The error object to be updated.

On input, the M-by-N matrix to factor. On output, the elements on and above the diagonal contain the MIN(M, N)-by-N upper trapezoidal matrix R (R is upper triangular if M >= N). The elements below the diagonal, along with the array tau, represent the orthogonal matrix Q as a product of elementary reflectors.

A MIN(M, N)-element array used to store the scalar factors of the elementary reflectors.

An optional input, that if provided, prevents any local memory allocation. If not provided, the memory required is allocated within. If provided, the length of the array must be at least olwork.

An optional output used to determine workspace size. If supplied, the routine determines the optimal size for work, and returns without performing any actual calculations.

The error object to be updated.

On input, the M-by-N matrix to factor. On output, the elements on and above the diagonal contain the MIN(M, N)-by-N upper trapezoidal matrix R (R is upper triangular if M >= N). The elements below the diagonal, along with the array tau, represent the orthogonal matrix Q as a product of elementary reflectors.

A MIN(M, N)-element array used to store the scalar factors of the elementary reflectors.

An optional input, that if provided, prevents any local memory allocation. If not provided, the memory required is allocated within. If provided, the length of the array must be at least olwork.

An optional output used to determine workspace size. If supplied, the routine determines the optimal size for work, and returns without performing any actual calculations.

The error object to be updated.

On input, the M-by-N matrix to factor. On output, the elements on and above the diagonal contain the MIN(M, N)-by-N upper trapezoidal matrix R (R is upper triangular if M >= N). The elements below the diagonal, along with the array tau, represent the orthogonal matrix Q as a product of elementary reflectors.

A MIN(M, N)-element array used to store the scalar factors of the elementary reflectors.

On input, an N-element array that if JPVT(I) .ne. 0, the I-th column of A is permuted to the front of A * P; if JPVT(I) = 0, the I-th column of A is a free column. On output, if JPVT(I) = K, then the I-th column of A * P was the K-th column of A.

An optional input, that if provided, prevents any local memory allocation. If not provided, the memory required is allocated within. If provided, the length of the array must be at least olwork.

An optional output used to determine workspace size. If supplied, the routine determines the optimal size for work, and returns without performing any actual calculations.

The error object to be updated.

On input, the M-by-N matrix to factor. On output, the elements on and above the diagonal contain the MIN(M, N)-by-N upper trapezoidal matrix R (R is upper triangular if M >= N). The elements below the diagonal, along with the array tau, represent the orthogonal matrix Q as a product of elementary reflectors.

A MIN(M, N)-element array used to store the scalar factors of the elementary reflectors.

On input, an N-element array that if JPVT(I) .ne. 0, the I-th column of A is permuted to the front of A * P; if JPVT(I) = 0, the I-th column of A is a free column. On output, if JPVT(I) = K, then the I-th column of A * P was the K-th column of A.

An optional input, that if provided, prevents any local memory allocation. If not provided, the memory required is allocated within. If provided, the length of the array must be at least olwork.

An optional output used to determine workspace size. If supplied, the routine determines the optimal size for work, and returns without performing any actual calculations.

An optional input, that if provided, prevents any local allocate of real-valued memory. If not provided, the memory required is allocated within. If provided, the length of the array must be at least 2*N.

The error object to be updated.

On input, the original M-by-K orthogonal matrix $Q$. On output, the updated matrix $Q_1$.

On input, the M-by-N matrix $R$. On output, the updated matrix $R_1$.

On input, the M-element $\vec{u}$ update vector. On output, the original content of the array is overwritten.

On input, the N-element $\vec{v}$ update vector. On output, the original content of the array is overwritten.

An optional argument that if supplied prevents local memory allocation. If provided, the array must have at least K elements.

The error object to be updated.

On input, the original M-by-K orthogonal matrix $Q$. On output, the updated matrix $Q_1$.

On input, the M-by-N matrix $R$. On output, the updated matrix $R_1$.

On input, the M-element $\vec{u}$ update vector. On output, the original content of the array is overwritten.

On input, the N-element $\vec{v}$ update vector. On output, the original content of the array is overwritten.

An optional argument that if supplied prevents local memory allocation. If provided, the array must have at least K elements.

An optional argument that if supplied prevents local memory allocation. If provided, the array must have at least K elements.

The error object to be updated.

On input, the M-by-N QR factored matrix as returned by qr_factor.
On output, the contents of this matrix are restored. Notice, M must be greater than or equal to N.

A MIN(M, N)-element array containing the scalar factors of the elementary reflectors as returned by qr_factor.

On input, the M-by-NRHS right-hand-side matrix. On output, the first N rows are overwritten by the solution matrix.

An optional input, that if provided, prevents any local memory allocation. If not provided, the memory required is allocated within. If provided, the length of the array must be at least olwork.

An optional output used to determine workspace size. If supplied, the routine determines the optimal size for work, and returns without performing any actual calculations.

The error object to be updated.

On input, the M-by-N QR factored matrix as returned by qr_factor.
On output, the contents of this matrix are restored. Notice, M must be greater than or equal to N.

A MIN(M, N)-element array containing the scalar factors of the elementary reflectors as returned by qr_factor.

On input, the M-by-NRHS right-hand-side matrix. On output, the first N rows are overwritten by the solution matrix.

An optional input, that if provided, prevents any local memory allocation. If not provided, the memory required is allocated within. If provided, the length of the array must be at least olwork.

An optional output used to determine workspace size. If supplied, the routine determines the optimal size for work, and returns without performing any actual calculations.

The error object to be updated.

On input, the M-by-N QR factored matrix as returned by qr_factor.
On output, the contents of this matrix are restored. Notice, M must be greater than or equal to N.

A MIN(M, N)-element array containing the scalar factors of the elementary reflectors as returned by qr_factor.

On input, the M-element right-hand-side vector. On output, the first N elements are overwritten with the solution vector.

An optional input, that if provided, prevents any local memory allocation. If not provided, the memory required is allocated within. If provided, the length of the array must be at least olwork.

An optional output used to determine workspace size. If supplied, the routine determines the optimal size for work, and returns without performing any actual calculations.

The error object to be updated.

On input, the M-by-N QR factored matrix as returned by qr_factor.
On output, the contents of this matrix are restored. Notice, M must be greater than or equal to N.

A MIN(M, N)-element array containing the scalar factors of the elementary reflectors as returned by qr_factor.

On input, the M-element right-hand-side vector. On output, the first N elements are overwritten with the solution vector.

An optional input, that if provided, prevents any local memory allocation. If not provided, the memory required is allocated within. If provided, the length of the array must be at least olwork.

An optional output used to determine workspace size. If supplied, the routine determines the optimal size for work, and returns without performing any actual calculations.

The error object to be updated.

On input, the M-by-N QR factored matrix as returned by qr_factor.
On output, the contents of this matrix are restored.

A MIN(M, N)-element array containing the scalar factors of the elementary reflectors as returned by qr_factor.

An N-element array, as output by qr_factor, used to track the column pivots.

On input, the MAX(M, N)-by-NRHS right-hand-side matrix. On output, the first N rows are overwritten by the solution matrix.

An optional input, that if provided, prevents any local memory allocation. If not provided, the memory required is allocated within. If provided, the length of the array must be at least olwork.

An optional output used to determine workspace size. If supplied, the routine determines the optimal size for work, and returns without performing any actual calculations.

The error object to be updated.

On input, the M-by-N QR factored matrix as returned by qr_factor.
On output, the contents of this matrix are restored.

A MIN(M, N)-element array containing the scalar factors of the elementary reflectors as returned by qr_factor.

An N-element array, as output by qr_factor, used to track the column pivots.

On input, the MAX(M, N)-by-NRHS right-hand-side matrix. On output, the first N rows are overwritten by the solution matrix.

An optional input, that if provided, prevents any local memory allocation. If not provided, the memory required is allocated within. If provided, the length of the array must be at least olwork.

An optional output used to determine workspace size. If supplied, the routine determines the optimal size for work, and returns without performing any actual calculations.

The error object to be updated.

On input, the M-by-N QR factored matrix as returned by qr_factor.
On output, the contents of this matrix are restored.

A MIN(M, N)-element array containing the scalar factors of the elementary reflectors as returned by qr_factor.

An N-element array, as output by qr_factor, used to track the column pivots.

On input, the MAX(M, N)-by-NRHS right-hand-side vector. On output, the first N rows are overwritten by the solution vector.

An optional input, that if provided, prevents any local memory allocation. If not provided, the memory required is allocated within. If provided, the length of the array must be at least olwork.

An optional output used to determine workspace size. If supplied, the routine determines the optimal size for work, and returns without performing any actual calculations.

The error object to be updated.

On input, the M-by-N QR factored matrix as returned by qr_factor.
On output, the contents of this matrix are restored.

A MIN(M, N)-element array containing the scalar factors of the elementary reflectors as returned by qr_factor.

An N-element array, as output by qr_factor, used to track the column pivots.

On input, the MAX(M, N)-by-NRHS right-hand-side vector. On output, the first N rows are overwritten by the solution vector.

An optional input, that if provided, prevents any local memory allocation. If not provided, the memory required is allocated within. If provided, the length of the array must be at least olwork.

An optional output used to determine workspace size. If supplied, the routine determines the optimal size for work, and returns without performing any actual calculations.

The error object to be updated.