Numerical Methods Library for OCTAVE

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3.4.1 Monomial basisP = ITPOL_MONOM(X,Y,x) computes the monomial basis interpolation of pointsdefined by x-coordinate X and y-coordinate Y. x can be a real vector, eachrow in the solution array P corresponds to a x-coordinate in the vector x.ParametersX abscissas of interpolated points.Y odinates of interpolated points.x can be a scalar or a vector of values.ReturnsP value of p(x).3.4.2 Lagrange interpolationP = LAGRANGE(X,Y,x) computes the polynomial Lagrange interpolation of pointsdefined by x-coordinate X and y-coordinate Y. x can be a real vector, eachrow in the solution array P corresponds to a x-coordinate in the vector x.ParametersX abscissas of interpolated points.Y odinates of interpolated points.x can be a scalar or a vector of values.ReturnsP value of P(x).3.4.3 Newton interpolationP = ITPOL_NEWT(X,Y,x) computes the polynomial Newton interpolation of pointsdefined by x-coordinate X and y-coordinate Y. x can be a real vector, eachrow in the solution array P corresponds to a x-coordinate in the vector x.ParametersX abscissas of interpolated points.Y odinates of interpolated points.x can be a scalar or a vector of values.ReturnsP value of p(x).3.5 Numerical integrationThe numerical approximation of definite integrals is known as numerical quadrature. This namederives from ancient methods for computing areas of curved figures, the most famous exampleof which is the problem of "squaring the circle" (finding a square having the same area as a13
given circle). In our case we wish to compute the area under a curve defined over an intervalon the real line. Thus, the quantity we wish to compute is of the formI( f ) =∫ baf (x)dx. (7)We will generally take the interval of integration to be finite, and we will assume for the mostpart that the integrand f is continuous and smooth. We will consider only briefly how to dealwith an infinite interval of integration or an integrand function that may have discontinuities orsingularities.Note that we seek a single number as an answer, not a function or a symbolic formula. Thisfeature distinguishes numerical quadrature from the solution of differential equations or theevaluation of indefinite integrals, as in elementary calculus and in many packages for symboliccomputation.An integral is, in effect, an infinite summation. It should come as no surprise that we willapproximate this infinite sum by a finite sum. Such a finite sum, in which the integrand functionis sampled at a finite number of points in the interval of integration, is called a quadraturerule. Our main object of study will be how to choose the sample points and how to weighttheir contributions to the quadrature formula so that we obtain a desired level of accuracy ata reasonable computational cost. For numerical quadrature, computational work is usuallymeasured by the number of evaluations of the integrand function that are required.3.5.1 Trapezoid’s ruleRES = INTE_TRAPEZ(FUN,A,B,N) computes an approximation of the integral ofthe function FUN via the trapezoid method (using N equispaced intervals).FUN accepts real scalar input x and returns a real scalar value. FUN canalso be an inline object.ParametersFUN integrated function.A,B FUN is integrated on [A,B].N number of subdivisions.ReturnsRES result of integration.3.5.2 Simpson’s ruleRES = INTE_SIMPSON(FUN,A,B,N) computes an approximation of the integral ofthe function FUN via the Simpson method (using N equispaced intervals). FUNaccepts real scalar input x and returns a real scalar value. FUN can alsobe an inline object.ParametersFUN integrated function.A,B FUN is integrated on [A,B].14
Page 3 and 4: 3.8.2 Implicit Euler . . . . . . .
Page 6 and 7: • Euler• Implicit Euler• Modi
Page 8 and 9: X0 initial guess.ReturnsX computed
Page 10 and 11: 3.2.1 Normal equation[X, RES] = NOR
Page 12 and 13: X0 initial point.ITMAX maximal numb
Page 16 and 17: N number of subdivisions.ReturnsRES
Page 18 and 19: 3.6.4 Orthogonal iteration[LAMBDA,
Page 20 and 21: H f (xk)s k = f (x k ) (12)for s k
Page 22 and 23: 3.8.1 Euler[TT,Y] = ODE_EULER(ODEFU
Page 24 and 25: condition Y0 using the fourth-order
Page 26 and 27: Figure 2: Shooting method for a two
Page 28 and 29: X equispaced grid of NX intervals.U
Page 30 and 31: ReturnsT grid of NT intervals.X gri
Page 32: A, BC, D rectangle (A,B)x(C,D) wher

Numerical Methods Library for OCTAVE

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