A reasonable representation (and one that fits well with the representations of other types) is to use single-element textures.
#SOLVING COMPLEX 4 EQUATION SYSTEMS HOW TO#
However, because many operations generate and work with single scalar values, we need to carefully choose how to represent such values on the GPU. At first glance, a representation of single floating-point values on the GPU might seem needless the values could just be given via uniform program parameters. In addition to representing vectors and matrices, the library needs to represent individual scalar floating-point values. In this section, we describe in more detail the internal representation of linear algebra operators in our GPU linear algebra library. Built upon efficient GPU representations of scalar values, vectors, and matrices, such a package can implement high-performance linear algebra operations such as vector-vector and matrix-vector operations. To solve linear PDEs on the GPU, we need a linear algebra package. Throughout this chapter, we show a variety of graphics effects that involve the solution of this PDE. Poisson's equation is of particular importance in physics, and its solution is frequently employed in computer graphics for the simulation of fluids and flow (as shown in Figure 44-10, later in the chapter).
We demonstrate the efficiency of our GPU solver using a particular PDE: the Poisson equation. The complete library, together with the "implicit water surface" demo (see Figure 44-9, later in the chapter), can be found on this book's CD.
Our system handles dense, banded, and general sparse matrices. The library provides routines for solving systems of linear equations, least-squares solutions of linear systems of equations, and standard operations on vector and matrix elements. We describe a C++ class hierarchy that allows easy and efficient use of the proposed operations. By means of these operations, implicit solvers for systems of algebraic equations can be implemented, thus enabling stable numerical simulation on programmable graphics hardware. Built upon efficient representations of vectors and matrices on the GPU, vector-vector and matrix-vector operations are implemented using fragment programs on DirectX 9-class hardware. In this chapter, we present a general framework for the computation of linear algebra operations on programmable graphics hardware. This system can then be solved using linear algebra operations. One of the basic methods to solve a PDE is to transform it into a large linear system of equations via discretization.
These techniques have a variety of applications in physics-based simulation and modeling, geometry processing, and image filtering, and they have been frequently employed in computer graphics to provide realistic simulation of real-world phenomena. The development of numerical techniques for solving partial differential equations (PDEs) is a traditional subject in applied mathematics. Technische Universität München 44.1 Overview A GPU Framework for Solving Systems of Linear Equations The CD content, including demos and content, is available on the web and for download.Ĭhapter 44. You can purchase a beautifully printed version of this book, and others in the series, at a 30% discount courtesy of InformIT and Addison-Wesley. GPU Gems 2 GPU Gems 2 is now available, right here, online.
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