Spline functions are universally recognized as highly effective tools in approximation theory, computer-aided geometric design, image analysis, and numerical analysis. The theory of univariate splines is well known but this text is the first comprehensive treatment of the analogous bivariate theory. A detailed mathematical treatment of polynomial splines on triangulations is outlined, providing a basis for developing practical methods for using splines in numerous application areas. The detailed treatment of the Bernstein-B??zier representation of polynomials will provide a valuable source for researchers and students in CAGD. Chapters on smooth macro-element spaces will allow engineers and scientists using the FEM method to solve partial differential equations numerically with new tools. Workers in the geosciences will find new tools for approximation and data fitting on the sphere. Ideal as a graduate text in approximation theory, and as a source book for courses in computer-aided geometric design or in finite-element methods.

This textbook presents a special solution of underdetermined linear systems where the number of nonzero entries in the solution is very small compared to the total number of entries. This is called sparse solution. As underdetermined linear systems can be very different, the authors explain how to compute a sparse solution by many approaches. Sparse Solutions of Underdetermined Linear Systems: Contains 72 algorithms for finding sparse solutions of underdetermined linear systems and their applications for matrix completion, graph clustering, and phase retrieval. Provides a detailed explanation of these algorithms including derivations and convergence analysis. Includes exercises for each chapter to help the reader understand the material. This textbook is appropriate for graduate students in math and applied math, computer science, statistics, data science, and engineering. Advisors and postdocs will also find the book of interest. It is appropriate for the following courses: Advanced Numerical Analysis, Special Topics on Numerical Analysis, Topics on Data Science, Topics on Numerical Optimization, and Topics on Approximation Theory.