Each page defines one weighted local least-squares problem. The columns of the weighted design matrix are normalized before diagnostics and regularization.
input: W_book : nn x 1 x N local weights G_book : nn x dim x N local basis values f_book : nn x numf × N local function values
output: c_book : dim × numf × N local coefficient vectors conds : N × 1 conditions of the solved normalized Gram systems info : struct additional regularization diagnostics
options: regularize mincond center amplification where activation starts maxcond center amplification where full correction is used targetcond inverse-amplification bound reached at full correction basis_degrees degree assigned to every basis column degree_laplace_shift shift in ell*(ell+shift); 1 on S2, 2 on S3 eval_vector basis values at the evaluation point (dim × 1 × N) numerical_cond_max condition cap for the nuisance block (default 1e10)
flags: centered_evaluation the evaluation vector is the first basis direction
The local systems are solved from the normalized normal equations, with one step of iterative refinement wherever no regularization is active. The evaluation direction is protected and only its orthogonal complement is regularized; one scalar parameter is chosen by bisection to reach the inverse-amplification target.
Syntax
[c_book, conds, info] = ...
Citing this page.
This page is part of the documentation of
MTEX, a free and open
source MATLAB toolbox for analyzing and modeling crystallographic textures.
It was written by The MTEX Developers and is published at
https://mtex-toolbox.github.io/solve_lsq_book_constsize.html.
If you use MTEX, or reuse text or figures from this page, in your research,
please cite
F. Bachmann, R. Hielscher, H. Schaeben: Texture Analysis with MTEX - Free and Open Source Software Toolbox, Solid State Phenomena 160 (2010), 63-68. 10.4028/www.scientific.net/SSP.160.63
BibTeX
@article{bachmann2010mtex,
author = {F. Bachmann and R. Hielscher and H. Schaeben},
title = {Texture Analysis with MTEX - Free and Open Source Software Toolbox},
journal = {Solid State Phenomena},
volume = {160},
pages = {63-68},
year = {2010},
doi = {10.4028/www.scientific.net/SSP.160.63},
url = {https://doi.org/10.4028/www.scientific.net/SSP.160.63}
}Other papers describing specific MTEX methods are listed under Publications — please cite the one that best fits your application. The MTEX source code is licensed under the GNU General Public License v2.0; the text and figures of this documentation are licensed under CC BY 4.0, which permits reuse — including by automated systems — provided The MTEX Developers and this page are credited.