# Formula Sheets (numerical methods)

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x iv+1  = f (x 1v , x 2v , x 3v , …, x nv )x iv+1  = f (x 1v+1 , x 2v+1 , …, x i-1v+1 , x iv , …, x nv )u(x,y) = f  1 (x,y) = 0If f(x l )*f(x r ) < 1 , x r  = x u v(x,y) = f  2 (x,y) = 0If f(x l )*f(x r ) > 1 , x r  = x l If f(x l )*f(x r ) < 1 , x r  = x u If f(x l )*f(x r ) > 1 , x r  = x l System of Linear Equationsh 0  = x 1 - x 0 h 1  = x 2 - x 1 b = ah 1  + δ 1 c = f(x 2 )  Ax  = b A  = LUUD  = bLx  = D if b is (+), b+sqrt…, otherwise b-sqrt  f(x) = a n x n  + a n-1 x n-1  + … + a 2 x 2  + a 1 x + a 0 a n  = 1assume r & sf  21  = a 21 /a 11 i = n-2 to 0f  31  = a 31 /a 11 b n  = a n c n  = b n f  32  = a' 32 /a' 22 b n-1  = a n-1  - rb n c n-1  = b n-1  - rc n b i  = a i  - rb i+1  - sb i+2 c i  = b i  - rc i+1  - sc i+2 c 2 Δr + c 3 Δs = b 1 c 1 Δr + c 2 Δs = b 0 r new  = r old  + Δr s new  = s old  + Δs f  21  = a 21 /a 11 f  32  = a 32 /a' 22 f  43  = a 43 /a'' 33 Convergence: M = N -1 P||M|| < 1for Jacobi Iter.InterpolationThomas AlgorithmLU DecompositionAbsolute ErrorRelative ErrorError RatioFixed Point IterationNewton's MethodGauss EliminationGauss-Jordan EliminationSecant MethodRegula Falsi / False PositionBisection / Half IntervalCholesky DecompositionJacobi IterationGauss-Seidel IterationNewton MethodGeneral SchemaSystem of Nonlinear EquationsApproximationMuller's MethodBairstow's MethodTaylor SeriesLagrange InterpolationSpline Interpolation  +1  =     U L  a 0  = f(x 0 )Curve Fittinge i  = f(x i ) - y i Numerical IntegrationRombergR k,j  = R k,j-1  + (R k,j-1  - R k-1,j-1 )/)4 j-1 - 1), k = j, j+1, … Finite Difference Method2D-Heat Equation 2κΔt/min((Δx) 2 ,(Δz) 2 ) ≤ 1 s x = κΔt/(Δx) 2 s z = κΔt/(Δz) 2 Solutions of ODEy' = f(x,y), y(x 0 )=y 0 y(x 0 +h) = y(x 0 ) + hy'(x 0 ) + (h 2 /2)y (x 0 ) + … y n+1  = y n  + h(k  1  + 4k  2  + k  3 )/6k  1  = f(x n , y n )k  2  = f(x n  + h/2, y n  + hk  1 /2)k  3  = f(x n  + h, y n  - hk  1  +2hk  2 )Implicit MethodTaylor Series ExpansionRKM 3rd Order2-Point Gaussian QuadratureExplicit Methodp n (x) = a 0  + a 1 (x - x 0 ) + a 2 (x - x 0 )(x - x 1 ) + … + a n-1 (x - x 0 )(x - x 1 ) … (x - x n-2 ) + a n (x - x 0 )(x - x 1 ) … (x - x n-2 )(x - x n-1 )Newton PolynomialLeast Squares LineGoodness of fit
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