linalg_least_squares Module

On input, the M-by-N matrix $A$. On output, if M is greater than or equal to N, the QR factorization of $A$ in the form provided by qr_factor; else, if M is less than N, the LQ factorization of $A$ as returned by lq_factor.

If the system is overdetermined, the M-by-NRHS matrix $B$; else, the matrix should be sized as N-by-NRHS with the first M rows containing $B$. On output, the first N rows will contain the solution matrix $X$.

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 $A$. On output, if M is greater than or equal to N, the QR factorization of $A$ in the form provided by qr_factor; else, if M is less than N, the LQ factorization of $A$ as returned by lq_factor.

If the system is overdetermined, the M-by-NRHS matrix $B$; else, the matrix should be sized as N-by-NRHS with the first M rows containing $B$. On output, the first N rows will contain the solution matrix $X$.

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 $A$. On output, if M is greater than or equal to N, the QR factorization of $A$ in the form provided by qr_factor; else, if M is less than N, the LQ factorization of $A$ as returned by lq_factor.

If the system is overdetermined, the M-element vector $\vec{b}$; else, the array should be sized as N-element with the first M elements containing $\vec{b}$. On output, the first N rows will contain the solution vector $\vec{x}$.

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 $A$. On output, if M is greater than or equal to N, the QR factorization of $A$ in the form provided by qr_factor; else, if M is less than N, the LQ factorization of $A$ as returned by lq_factor.

If the system is overdetermined, the M-element vector $\vec{b}$; else, the array should be sized as N-element with the first M elements containing $\vec{b}$. On output, the first N rows will contain the solution vector $\vec{x}$.

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 $A$. On output, the matrix is overwritten by its orthogonal factorization.

If the system is overdetermined, the M-by-NRHS matrix $B$; else, the matrix should be sized as N-by-NRHS with the first M rows containing $B$. On output, the first N rows will contain the solution matrix $X$.

An optional input that on input, an N-element array that if IPVT(I) .ne. 0, the I-th column of A is permuted to the front of A * P; if IPVT(I) = 0, the I-th column of A is a free column. On output, if IPVT(I) = K, then the I-th column of A * P was the K-th column of A. If not supplied, memory is allocated internally, and IPVT is set to all zeros such that all columns are treated as free.

An optional output, that if provided, will return the rank 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 $A$. On output, the matrix is overwritten by its orthogonal factorization.

If the system is overdetermined, the M-by-NRHS matrix $B$; else, the matrix should be sized as N-by-NRHS with the first M rows containing $B$. On output, the first N rows will contain the solution matrix $X$.

An optional input that on input, an N-element array that if IPVT(I) .ne. 0, the I-th column of A is permuted to the front of A * P; if IPVT(I) = 0, the I-th column of A is a free column. On output, if IPVT(I) = K, then the I-th column of A * P was the K-th column of A. If not supplied, memory is allocated internally, and IPVT is set to all zeros such that all columns are treated as free.

An optional output, that if provided, will return the rank 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 memory allocation for real-valued workspaces. 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 M-by-N matrix $A$. On output, the matrix is overwritten by its orthogonal factorization.

If the system is overdetermined, the M-element vector $\vec{b}$; else, the array should be sized as N-element with the first M elements containing $\vec{b}$. On output, the first N rows will contain the solution vector $\vec{x}$.

An optional input that on input, an N-element array that if IPVT(I) .ne. 0, the I-th column of A is permuted to the front of A * P; if IPVT(I) = 0, the I-th column of A is a free column. On output, if IPVT(I) = K, then the I-th column of A * P was the K-th column of A. If not supplied, memory is allocated internally, and IPVT is set to all zeros such that all columns are treated as free.

An optional output, that if provided, will return the rank 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 $A$. On output, the matrix is overwritten by its orthogonal factorization.

If the system is overdetermined, the M-element vector $\vec{b}$; else, the array should be sized as N-element with the first M elements containing $\vec{b}$. On output, the first N rows will contain the solution vector $\vec{x}$.

An optional input that on input, an N-element array that if IPVT(I) .ne. 0, the I-th column of A is permuted to the front of A * P; if IPVT(I) = 0, the I-th column of A is a free column. On output, if IPVT(I) = K, then the I-th column of A * P was the K-th column of A. If not supplied, memory is allocated internally, and IPVT is set to all zeros such that all columns are treated as free.

An optional output, that if provided, will return the rank 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 memory allocation for real-valued workspaces. 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 M-by-N matrix $A$. On output, the matrix is overwritten.

If the system is overdetermined, the M-by-NRHS matrix $B$; else, the matrix should be sized as N-by-NRHS with the first M rows containing $B$. On output, the first N rows will contain the solution matrix $X$.

An optional MIN(M, N)-element array that on output contains the singular values of $A$ in descending order. Notice, the condition number of $A$ can be determined by S(1) / S(MIN(M, N)).

An optional output, that if provided, will return the rank 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 $A$. On output, the matrix is overwritten.

If the system is overdetermined, the M-by-NRHS matrix $B$; else, the matrix should be sized as N-by-NRHS with the first M rows containing $B$. On output, the first N rows will contain the solution matrix $X$.

An optional MIN(M, N)-element array that on output contains the singular values of $A$ in descending order. Notice, the condition number of $A$ can be determined by S(1) / S(MIN(M, N)).

An optional output, that if provided, will return the rank 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 memory allocation for real-valued workspaces. If not provided, the memory required is allocated within. If provided, the length of the array must be at least 5 * MIN(M, N).

The error object to be updated.

On input, the M-by-N matrix $A$. On output, the matrix is overwritten.

If the system is overdetermined, the M-element vector $\vec{b}$; else, the array should be sized as N-element with the first M elements containing $\vec{b}$. On output, the first N rows will contain the solution vector $\vec{x}$.

An optional MIN(M, N)-element array that on output contains the singular values of $A$ in descending order. Notice, the condition number of $A$ can be determined by S(1) / S(MIN(M, N)).

An optional output, that if provided, will return the rank 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 $A$. On output, the matrix is overwritten.

If the system is overdetermined, the M-element vector $\vec{b}$; else, the array should be sized as N-element with the first M elements containing $\vec{b}$. On output, the first N rows will contain the solution vector $\vec{x}$.

An optional MIN(M, N)-element array that on output contains the singular values of $A$ in descending order. Notice, the condition number of $A$ can be determined by S(1) / S(MIN(M, N)).

An optional output, that if provided, will return the rank 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 memory allocation for real-valued workspaces. If not provided, the memory required is allocated within. If provided, the length of the array must be at least 5 * MIN(M, N).

The error object to be updated.