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 $Q R = [Q_1, Q_2] [R1 0]^T$.

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 $Q R = [Q_1, Q_2] [R1 0]^T$.

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.

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.

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.

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

The M-by-N matrix $C$.

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.

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

The M-by-N matrix $C$.

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.

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

The M-element vector $\vec{c}$.

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$A$.

The M-element vector $\vec{c}$.

The M-by-N matrix to factor.

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

The M-by-N factored matrix stored such that 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.

The M-by-M orthogonal matrix $Q$.

The M-by-N upper trapezoidal matrix $R$.

The M-by-N matrix to factor.

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

The M-by-N factored matrix stored such that 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.

The M-by-M orthogonal matrix $Q$.

The M-by-N upper trapezoidal matrix $R$.

The M-by-N matrix to factor.

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.

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

The M-by-N factored matrix stored such that 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.

The M-by-M orthogonal matrix $Q$.

The M-by-N upper trapezoidal matrix $R$.

The N-by-N column-pivot tracking matrix $P$ where $A P = Q R$.

The M-by-N matrix to factor.

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.

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

The M-by-N factored matrix stored such that 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.

The M-by-M orthogonal matrix $Q$.

The M-by-N upper trapezoidal matrix $R$.

The N-by-N column-pivot tracking matrix $P$ where $A P = Q R$.

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 $R_1$.

The M-element $\vec{u}$ update vector.

The N-element $\vec{v}$ update vector.

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.

The M-by-N QR factored matrix as returned by qr_factor.

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

The M-by-NRHS right-hand-side matrix.

The M-by-N QR factored matrix as returned by qr_factor.

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

The M-by-NRHS right-hand-side matrix.

The M-by-N QR factored matrix as returned by qr_factor.

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

The M-element right-hand-side vector.

The M-by-N QR factored matrix as returned by qr_factor.

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

The M-element right-hand-side vector.

The M-by-N QR factored matrix as returned by qr_factor.

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.

The MAX(M, N)-by-NRHS right-hand-side matrix.

The M-by-N QR factored matrix as returned by qr_factor.

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.

The M-by-NRHS right-hand-side matrix.

The M-by-N QR factored matrix as returned by qr_factor.

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.

The M-element right-hand-side vector.

The M-by-N QR factored matrix as returned by qr_factor.

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.

The M-element right-hand-side vector.