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Regula falsiTake two starting values x 1 and x 2with f (x 1 ) < 0 and f (x 2 ) > 0.Approximate f by a straight line betweenf (x 1 ) and f (x 2 ) and calculate its root as:x m = (f((x 1 ) x 2 – f (x 2 ) x 1 ) / (f (x 1 ) – f (x 2 )).If sign(f (x m )) = sign(f (x 1 )), thenreplace x 1 by x m otherwise replace x 2 by x m .Regula falsi7980
N-dimensionalequationsBex a N-dimensionalvector.System of N coupled equations:f( x)0corresponding to the N-dimensionaloptimization problem: ( x F ) 081N-dimensional Newton methodJDefine the Jacobi matrix:i,j fi( x)( x) xjMust be non-singularandalso well-conditionedfor numerical inversion. x x J f ( x )1n 1 n n82
Page 1 and 2: Monte Carlo MethodsNicholas Constan
Page 3 and 4: Calculation of π1x 2 + y 2 = 1 124
Page 5 and 6: Error of integrationconventional me
Page 7 and 8: Not smooth integralsg(x)ax ineed mo
Page 9 and 10: M(RT) 2 algorithmN.C. Metropolis, A
Page 11 and 12: Detailed balancedp( X , t)dt pY ( )
Page 13 and 14: The Ising ModelBinary variables:i1,
Page 15 and 16: Binary mixtures(lattice gas)Conside
Page 17 and 18: Ising interface+++ +NH E J ij+i, j
Page 19 and 20: Ising interfaceAdd next-nearest nea
Page 21 and 22: Irreversible growth• T. Vicsek,
Page 23 and 24: Random deposition with surface diff
Page 25 and 26: Ballistic depositionParticles fall
Page 27 and 28: DLA clustersfractaldimensiond f = 1
Page 29 and 30: Dielectric breakdown57Dielectric br
Page 31 and 32: Dielectric breakdown modelη = 0.16
Page 33 and 34: Simulated Annealing (SA)S. Kirckpat
Page 35 and 36: Slow coolingT 0T 1T 20Different coo
Page 37 and 38: Solving equationsFinding the soluti
Page 39: Bisection methodTake two starting v
Page 43 and 44: N-dimensionalsecant methodIf the de
Page 45 and 46: Euler methodStart with y 0 and iter
Page 47 and 48: The order of a methodIf the error a
Page 49 and 50: Fourth order Runge - Kuttak1tf( y ,
Page 51 and 52: Example RK4Butcher array:its stage
Page 53 and 54: Example q = p = 2y ny t ( 1 )
Page 55 and 56: Adaptive time-stepΔtδ expectedDef
Page 57 and 58: Richardson extrapolationCalculate f
Page 59 and 60: 3th order predictor- correctoruse e
Page 61 and 62: Comparison of methodsΔtfor fixed n

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