Moving Least Squares fits a small weighted polynomial around every point x. Each node I has a weight function w_I(x) that is zero outside its support domain of radius d_I = αh, where h is the nodal spacing and α is the dilation parameter. The approximation and its shape functions are

• α too small. Some points see fewer than m nodes, A(x) becomes singular and the approximation breaks down. Near the boundary the support is one-sided, so on a regular grid the quadratic basis needs α > 2 there, but only α > 1.5 in the interior.
• α too large. Every point averages over many nodes, the fit over-smooths, locality is lost and cond A(x) grows.
• In between lies an α* that minimises the error. It depends on the basis, the weight function, the node layout and the function itself, so the sweep is recomputed for your current settings.
• MLS does not interpolate. In general u^h(x_I) ≠ u_I. This is why the Element-Free Galerkin method needs special treatment of essential boundary conditions, such as Lagrange multipliers, penalty, or coupling with finite elements.

Implementation: shifted and scaled basis p(s) = [1, s, s²] with s = (x_I − x)/h, which keeps A(x) well conditioned. Errors are relative L2 norms over 401 sample points on [0, 1]; du/dx is obtained by second-order finite differences of u^h.

1. Choose f = 1 + 2x − 3x² with the quadratic basis. The error drops to machine precision for every α that keeps A(x) non-singular. This is the reproducing, or consistency, property.
2. Keep that function and switch to the linear basis. The error now depends on α. Where is α*?
3. With the quadratic basis, slide α down from 2.5 towards 1.5. Where does the singular region appear first?
4. Pick the tanh function and switch the sweep between u and du/dx. Is α* the same for both?
5. Switch to irregular nodes and press “New layout” a few times. How do α* and the singular region move?
6. Double the number of nodes and read the error. How fast does it fall with h for each basis?

Moving Least Squares memadankan polinomial berpemberat yang kecil di sekitar setiap titik x. Setiap nod I mempunyai weight function w_I(x) yang sifar di luar support domain berjejari d_I = αh, dengan h ialah jarak antara nod dan α ialah dilation parameter. Approximation dan shape functions-nya ialah

• α terlalu kecil. Sesetengah titik melihat kurang daripada m nod, A(x) menjadi singular dan approximation gagal. Berhampiran sempadan support hanya sebelah, jadi pada grid seragam basis kuadratik memerlukan α > 2 di situ, tetapi hanya α > 1.5 di bahagian dalam.
• α terlalu besar. Setiap titik mempuratakan banyak nod, padanan menjadi terlalu licin, sifat setempat hilang dan cond A(x) meningkat.
• Di antaranya terdapat α* yang meminimumkan ralat. Ia bergantung pada basis, weight function, susunan nod dan fungsi itu sendiri, jadi sweep dikira semula untuk tetapan semasa anda.
• MLS tidak menginterpolasi. Secara umum u^h(x_I) ≠ u_I. Inilah sebabnya Element-Free Galerkin method memerlukan rawatan khas bagi essential boundary conditions, seperti Lagrange multipliers, penalty, atau gandingan dengan finite elements.

Pelaksanaan: basis teranjak dan berskala p(s) = [1, s, s²] dengan s = (x_I − x)/h, yang mengekalkan A(x) berkeadaan baik. Ralat ialah norma relatif L2 atas 401 titik sampel pada [0, 1]; du/dx diperoleh melalui finite differences peringkat kedua bagi u^h.

1. Pilih f = 1 + 2x − 3x² dengan basis kuadratik. Ralat jatuh ke ketepatan mesin bagi setiap α yang mengekalkan A(x) tidak singular. Inilah sifat reproducing, atau consistency.
2. Kekalkan fungsi itu dan tukar ke basis linear. Ralat kini bergantung pada α. Di manakah α*?
3. Dengan basis kuadratik, gerakkan α turun dari 2.5 ke arah 1.5. Di manakah kawasan singular mula-mula muncul?
4. Pilih fungsi tanh dan tukar sweep antara u dan du/dx. Adakah α* sama bagi kedua-duanya?
5. Tukar ke nod tak seragam dan tekan “Susunan baharu” beberapa kali. Bagaimana α* dan kawasan singular berubah?
6. Gandakan bilangan nod dan baca ralatnya. Seberapa pantas ia jatuh dengan h bagi setiap basis?