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230 CHAPTER 8. FINITE WORLD OF COMPUTERS 8.2.5 Finite Element Integration In the theory of finite elements [?, ?], a continuous function space is projected onto a finite function space P k , where the space P k is the space of polynomials up to the total order of k. For many special cases, finite element integrals can be computed manually and added into the source code of an application. This results in excellent run time performance but lacks flexibility. For more general cases, e.g., general coefficients, they must be computed by numerical integration at run-time. To prevent an ill-conditioned system matrix, orthogonal polynomials have to be chosen as numerical integration weights. One possible type of polynomial is a normalized Legendre polynomial [?]. Coefficients for such a polynomial Pk of order k can be efficiently evaluated by using the recursion procedure: P0(x) = 1 (8.1) P1(x) = x 2j − 1 Pk(x) = x Pk−1(x) − j j − 1 j Pk−2(x) k ≥ 2 To use arbitrary p-finite elements (polynomial order [?, ?]) the numerical coefficients have to be calculated either manually and inserted into the source code or determined numerically at run time. The polynomial library presented here is then used to store manually pre-calculated integration tables at compile time (order 1-5). If the user requires higher order finite elements, numerical coefficients are calculated at run time to any order. 8.3 A Loop and More One of the important parts in computer science is repetation. A computer was made to do exactly like this, programmable operations and repetations. To give a simple example, a for loop is expressed by: for (long i = 0; i < max counter; ++i) {} To give a real application of this concept, integration is used. � b a f(x)dx Several approximation schemes are also available: � b a � b a f(x)dx ≈ f(x)dx ≈ b − a 6 f(a) + f(b) 2 � f(a) + 4f · (b − a) � a + b 2 � � + f(b)
8.4. THE OTHER WAY AROUND 231 As can be seen, this is a very coarse approximation, but the main issue persists. The known continuos integration is not possible inside the computer, but the concept of numerical integration is possible. This means the constraint of a finite dx is replaced by a ∆x and the � is replaced by a finite sum � i=0;i
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Technische Universität Dresden Fak
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Contents I Understanding C++ 7 Intr
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CONTENTS 5 10.5 Unix and Linux . .
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Part I Understanding C ++ 7
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10 C ++ was not a reliable computer
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12 CHAPTER 1. GOOD AND BAD SCIENTIF
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20 CHAPTER 2. C++ BASICS • std::c
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22 CHAPTER 2. C++ BASICS In the fir
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24 CHAPTER 2. C++ BASICS int main (
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26 CHAPTER 2. C++ BASICS Operator A
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28 CHAPTER 2. C++ BASICS The bitwis
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30 CHAPTER 2. C++ BASICS cast (type
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32 CHAPTER 2. C++ BASICS complicate
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34 CHAPTER 2. C++ BASICS } else if
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36 CHAPTER 2. C++ BASICS eps/= 2.0;
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38 CHAPTER 2. C++ BASICS for (...;
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40 CHAPTER 2. C++ BASICS 2.6.1 Inli
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42 CHAPTER 2. C++ BASICS To make su
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44 CHAPTER 2. C++ BASICS float divi
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46 CHAPTER 2. C++ BASICS The first
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48 CHAPTER 2. C++ BASICS #ifndef at
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50 CHAPTER 2. C++ BASICS float A[7]
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52 CHAPTER 2. C++ BASICS Encapsulat
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54 CHAPTER 2. C++ BASICS As a pract
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56 CHAPTER 2. C++ BASICS A function
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58 CHAPTER 2. C++ BASICS Now that t
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60 CHAPTER 2. C++ BASICS we need sm
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62 CHAPTER 2. C++ BASICS 2.12 Exerc
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64 CHAPTER 2. C++ BASICS 2.13 Opera
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66 CHAPTER 3. CLASSES apply symm bl
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68 CHAPTER 3. CLASSES int main() {
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70 CHAPTER 3. CLASSES class solver
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72 CHAPTER 3. CLASSES • Compilers
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74 CHAPTER 3. CLASSES a real number
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76 CHAPTER 3. CLASSES } return ∗t
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78 CHAPTER 3. CLASSES This mechanis
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80 CHAPTER 3. CLASSES One could not
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82 CHAPTER 3. CLASSES class matrix
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84 CHAPTER 3. CLASSES Approach 3: R
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86 CHAPTER 3. CLASSES of the decrem
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88 CHAPTER 3. CLASSES There are two
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90 CHAPTER 4. GENERIC PROGRAMMING }
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92 CHAPTER 4. GENERIC PROGRAMMING c
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94 CHAPTER 4. GENERIC PROGRAMMING A
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96 CHAPTER 4. GENERIC PROGRAMMING v
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98 CHAPTER 4. GENERIC PROGRAMMING c
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100 CHAPTER 4. GENERIC PROGRAMMING
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150 CHAPTER 5. META-PROGRAMMING 5.3
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152 CHAPTER 5. META-PROGRAMMING •
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156 CHAPTER 5. META-PROGRAMMING };
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162 CHAPTER 5. META-PROGRAMMING num
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168 CHAPTER 5. META-PROGRAMMING The
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Page 237 and 238: 10.2. DEBUGGING 237 T& glas::contin
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Page 247 and 248: 11.3. BOOST.BINDINGS 247 #include
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14.7. SEQUENTIAL NOISE REDUCTION OF
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Programmierprojekte Kapitel 15 Die
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15.6. MATRIX-SKALIERUNG 287 Siehe h
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15.10. ANWENDUNG MTL4 AUF TYPEN MIT
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Acknowledgement Chapter 16 Special
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Bibliography [AG04] David Abrahams
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