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This function allocates a workspace for solving large linear least squares systems. The least squares matrix X has p columns, but may have any number of rows. The parameter T specifies the method to be used for solving the large least squares system and may be selected from the following choices
This specifies the normal equations approach for solving the least squares system. This method is suitable in cases where performance is critical and it is known that the least squares matrix X is well conditioned. The size of this workspace is O(p^2).
This specifies the sequential Tall Skinny QR (TSQR) approach for solving the least squares system. This method is a good general purpose choice for large systems, but requires about twice as many operations as the normal equations method for n >> p. The size of this workspace is O(p^2).
This function frees the memory associated with the workspace w.
This function returns a string pointer to the name of the multilarge solver.
This function resets the workspace w so it can begin to accumulate a new least squares system.
These functions define a regularization matrix
L = diag(l_0,l_1,...,l_{p-1}).
The diagonal matrix element l_i is provided by the
ith element of the input vector L.
The block (X,y) is converted to standard form and
the parameters (\tilde{X},\tilde{y}) are stored in Xs
and ys on output. Xs and ys have the same dimensions as
X and y. Optional data weights may be supplied in the
vector w. In order to apply this transformation,
L^{-1} must exist and so none of the l_i
may be zero. After the standard form system has been solved,
use gsl_multilarge_linear_genform1
to recover the original solution vector.
It is allowed to have X = Xs and y = ys for an in-place transform.
This function calculates the QR decomposition of the m-by-p regularization matrix
L. L must have m \ge p. On output,
the Householder scalars are stored in the vector tau of size p.
These outputs will be used by gsl_multilarge_linear_wstdform2
to complete the
transformation to standard form.
These functions convert a block of rows (X,y,w) to standard
form (\tilde{X},\tilde{y}) which are stored in Xs and ys
respectively. X, y, and w must all have the same number of rows.
The m-by-p regularization matrix L is specified by the inputs
LQR and Ltau, which are outputs from gsl_multilarge_linear_L_decomp
.
Xs and ys have the same dimensions as X and y. After the
standard form system has been solved, use gsl_multilarge_linear_genform2
to
recover the original solution vector. Optional data weights may be supplied in the
vector w, where W = diag(w).
This function accumulates the standard form block (X,y) into the current least squares system. X and y have the same number of rows, which can be arbitrary. X must have p columns. For the TSQR method, X and y are destroyed on output. For the normal equations method, they are both unchanged.
After all blocks (X_i,y_i) have been accumulated into the large least squares system, this function will compute the solution vector which is stored in c on output. The regularization parameter \lambda is provided in lambda. On output, rnorm contains the residual norm ||y - X c||_W and snorm contains the solution norm ||L c||.
After a regularized system has been solved with L = diag(\l_0,\l_1,...,\l_{p-1}), this function backtransforms the standard form solution vector cs to recover the solution vector of the original problem c. The diagonal matrix elements l_i are provided in the vector L. It is allowed to have c = cs for an in-place transform.
After a regularized system has been solved with a regularization matrix L, specified by (LQR,Ltau), this function backtransforms the standard form solution cs to recover the solution vector of the original problem, which is stored in c, of length p.
This function computes the reciprocal condition number, stored in rcond, of the least squares matrix after it has been accumulated into the workspace work. For the TSQR algorithm, this is accomplished by calculating the SVD of the R factor, which has the same singular values as the matrix X. This routine is currently not implemented for the normal equations method.
Previous: Large Linear Systems Solution Steps, Up: Large Linear Systems [Index]