#### 38.6.4 Large Dense Linear Least Squares Routines

Function: gsl_multilarge_linear_workspace * gsl_multilarge_linear_alloc (const gsl_multilarge_linear_type * T, const size_t p)

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

Multilarge type: gsl_multilarge_linear_normal

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).

Multilarge type: gsl_multilarge_linear_tsqr

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).

Function: void gsl_multilarge_linear_free (gsl_multilarge_linear_workspace * w)

This function frees the memory associated with the workspace w.

Function: const char * gsl_multilarge_linear_name (gsl_multilarge_linear_workspace * w)

This function returns a string pointer to the name of the multilarge solver.

Function: int gsl_multilarge_linear_reset (gsl_multilarge_linear_workspace * w)

This function resets the workspace w so it can begin to accumulate a new least squares system.

Function: int gsl_multilarge_linear_stdform1 (const gsl_vector * L, const gsl_matrix * X, const gsl_vector * y, gsl_matrix * Xs, gsl_vector * ys, gsl_multilarge_linear_workspace * work)
Function: int gsl_multilarge_linear_wstdform1 (const gsl_vector * L, const gsl_matrix * X, const gsl_vector * w, const gsl_vector * y, gsl_matrix * Xs, gsl_vector * ys, gsl_multilarge_linear_workspace * work)

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.

Function: int gsl_multilarge_linear_L_decomp (gsl_matrix * L, gsl_vector * tau)

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.

Function: int gsl_multilarge_linear_stdform2 (const gsl_matrix * LQR, const gsl_vector * Ltau, const gsl_matrix * X, const gsl_vector * y, gsl_matrix * Xs, gsl_vector * ys, gsl_multilarge_linear_workspace * work)
Function: int gsl_multilarge_linear_wstdform2 (const gsl_matrix * LQR, const gsl_vector * Ltau, const gsl_matrix * X, const gsl_vector * w, const gsl_vector * y, gsl_matrix * Xs, gsl_vector * ys, gsl_multilarge_linear_workspace * work)

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).

Function: int gsl_multilarge_linear_accumulate (gsl_matrix * X, gsl_vector * y, gsl_multilarge_linear_workspace * 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.

Function: int gsl_multilarge_linear_solve (const double lambda, gsl_vector * c, double * rnorm, double * snorm, gsl_multilarge_linear_workspace * w)

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||.

Function: int gsl_multilarge_linear_genform1 (const gsl_vector * L, const gsl_vector * cs, gsl_vector * c, gsl_multilarge_linear_workspace * work)

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.

Function: int gsl_multilarge_linear_genform2 (const gsl_matrix * LQR, const gsl_vector * Ltau, const gsl_vector * cs, gsl_vector * c, gsl_multilarge_linear_workspace * work)

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.

Function: int gsl_multilarge_linear_lcurve (gsl_vector * reg_param, gsl_vector * rho, gsl_vector * eta, gsl_multilarge_linear_workspace * work)

This function computes the L-curve for a large least squares system after it has been fully accumulated into the workspace work. The output vectors reg_param, rho, and eta must all be the same size, and will contain the regularization parameters \lambda_i, residual norms ||y - X c_i||, and solution norms || L c_i || which compose the L-curve, where c_i is the regularized solution vector corresponding to \lambda_i. The user may determine the number of points on the L-curve by adjusting the size of these input arrays. For the TSQR method, the regularization parameters \lambda_i are estimated from the singular values of the triangular R factor. For the normal equations method, they are estimated from the eigenvalues of the X^T X matrix.

Function: int gsl_multilarge_linear_rcond (double * rcond, gsl_multilarge_linear_workspace * work)

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. For the normal equations method, this is done by computing the eigenvalues of X^T X, which could be inaccurate for ill-conditioned matrices X.