By Philipp Scherer

ISBN-10: 3642139892

ISBN-13: 9783642139895

ISBN-10: 3642139906

ISBN-13: 9783642139901

This booklet encapsulates the insurance for a two-semester path in computational physics. the 1st half introduces the elemental numerical equipment whereas omitting mathematical proofs yet demonstrating the algorithms in terms of a variety of laptop experiments. the second one half focuses on simulation of classical and quantum platforms with instructive examples spanning many fields in physics, from a classical rotor to a quantum bit. All application examples are discovered as Java applets able to run on your browser and don't require any programming talents.

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Example text

1) Ax = b. 2) an1 x1 + · · · + ann xn = bn or shortly If the dimension of the system is not too large standard methods like Gaussian elimination are sufficient. However, many applications involve very large dimensions and require special iterative methods. 1 Gaussian Elimination Method A series of linear combinations of the equations transforms the matrix A into an upper triangular matrix. n ⎞ ⎞ ⎛ at1 at1 ⎜ at ⎟ ⎜ a2 − l21 at ⎟ ⎜ 2⎟ ⎜ 1⎟ ⎟, ⎜ .. ⎟ → ⎜ .. ⎠ ⎝ . ⎠ ⎝ . J. 1007/978-3-642-13990-1_5, C Springer-Verlag Berlin Heidelberg 2010 47 48 5 Systems of Inhomogeneous Linear Equations ⎛ ⎞ 1 ⎜ −l21 1 ⎟ ⎜ ⎟ ⎜ −l31 1 ⎟ L1 = ⎜ ⎟ ⎜ ..

N = 2 gives Simpson’s rule 2h f0 + 4 f1 + f2 . 11) 19 f 0 +75 f 1 +50 f 2 +50 f 3 +75 f 4 +19 f 5 288 41 f 0 +216 f 1 +27 f 2 +272 f 3 +27 f 4 +216 f 5 +41 f 6 840 Weddle rule For even larger n negative weight factors appear and the formulas are not numerically stable. 2 Newton–Cotes Expressions for an Open Interval If the function has a singularity at the end of the interval, it is more convenient to compute the integral from only interior points xi = a + i h i = 1, 2, . . , N b−a . 12) a+b .

49) Problems 45 √ 1 3 x2 1 w1 = L 1 dx = − = 1, − x 2 2 3 −1 −1 √ 1 1 1 3 x2 L 2 dx = = 1. 51) −1 This gives the integral rule 1 −1 f (x)dx ≈ f − 1 3 + f 1 . 53) and find the approximation b f (x)dx = a b−a ≈ 2 f 1 −1 f a+b b−a + u 2 2 a+b b−a − 2 2 1 3 + f b−a du 2 a+b b−a + 2 2 1 3 . 77459 . . , x2 = 0. 55) Besides these Gaussian (Legendre) expressions further integral rules can be obtained by using other sets of orthogonal polynomials, for instance Laguerre, Hermite, or Jacobi polynomials. 1 Romberg integration Use the trapezoidal rule T (h) = h 1 1 f (a) + f (a + h) + · · · + f (b − h) + f (b) = 2 2 with the step sequence b a f (x)dx + · · · 46 4 Numerical Integration hi = h0 2i and calculate the elements of the triangular matrix T (i, 0) = T (h i ) T (i, k) = T (i + 1, k − 1) + T (i, k − 1) − T (i + 1, k − 1) 1− h i2 2 h i+k to obtain the approximations T01 = P01 , T02 = P012 , T03 = P0123 , .

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Computational Physics: Simulation of Classical and Quantum Systems by Philipp Scherer


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