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- Question : 1 - Write a computer program for Lagrange interpolation (you may want to use the Numerical Recipes subroutine polint or interp1 of MATLAB). Test your program by verifying that P(0.7) = ?0.226 in Example 1.1. (a) Using the data of Example 1.1, find the interpolated value at x = 0.9. (b) Use Runge
- Question : 2 - Derive an expression for the derivative of a Lagrange polynomial of order n at a point x between the data points.
- Question : 3 - Showthat if parabolic run-out conditions are used for cubic spline interpolation, then the interpolating polynomials in the first and last intervals are indeed parabolas.
- Question : 4 - An operationally simpler spline is the so-called quadratic spline. Interpolation is carried out by piecewise quadratics. (a) What are the suitable joint conditions for quadratic spline? (b) Show how the coefficients of the spline are obtained. What are suitable end conditions? (c) Compare the required computational efforts for quadratic and cubic splines.
- Question : 5 - Consider a set of n + 1 data points (x0, f0), . . . , (xn, fn), equally spaced with xi+1 ? xi = h. Discuss how cubic splines can be used to obtain a numerical approximation for the first derivative f at these data points. Give a detailed account of the required steps. You should obtain formulas for the numerical derivative at the data points x0, . . . , xn and explain how to calculate the terms in the formulas.
- Question : 6 - Tension splines can be used if the interpolating spline wiggles too much. In this case, the equation governing the position of the plastic ruler in between the data points is y(iv) ? ?2 y = 0 where ? is the tension parameter. If we denote gi (x) as the interpolating tension spline in the interval xi ? x ? xi+1, then g i (x) ? ?2gi (x) is a straight line in (a) Verify that for ? = 0, the cubic spline is recovered, and ? ??leads to linear interpolation. (b) Derive the equation for tension spline interpolation, i.e., the expression for gi (x).
- Question : 7 - The tuition for 12 units at St. Anford University has been increasing from 1998 to 2008 as shown in the table below: Year Tuition per year 1998 $21,300 1999 $23,057 2000 $24,441 2001 $25,917 2002 $27,204 2003 $28,564 2004 $29,847 2005 $31,200 2006 $32,994 2007 $34,800 2008 $36,030 (a) Plot the given data points and intuitively interpolate (draw) a smooth curve through them. (b) Interpolate the data with the Lagrange polynomial. Plot the polynomial and the data points. Use the polynomial to predict the tuition in 2010. This is an extrapolation problem; discuss the utility of Lagrange polynomials for extrapolation. (c) Repeat (b) with a cubic spline interpolation and compare your results.
- Question : 8 - The concentration of a certain toxin in a system of lakes downwind of an industrial area has been monitored very accurately at intervals from 1993 to 2007 as shown in the table below. It is believed that the concentration has varied smoothly between these data points. Year Toxin Concentration 1993 12.0 1995 12.7 1997 13.0 1999 15.2 2001 18.2 2003 19.8 2005 24.1 2007 28.1 2009 ??? (a) Interpolate the data with the Lagrange polynomial. Plot the polynomial and the data points. Use the polynomial to predict the condition of the lakes in 2009. Discuss this prediction. (b) Interpolation may also be used to fill
- Question : 9 - Consider a piecewise Lagrange polynomial that interpolates between three points at a time. Let a typical set of consecutive three points be xi?1, xi, and xi+1. Derive differentiation formulas for the first and second derivatives at xi. Simplify these expressions for uniformly spaced data with = xi+1 ? xi . You have just derived finite difference formulas for discrete data, which are discussed in the next chapter.
- Question : 10 - Consider a function f defined on a set of N + 1 discrete points x0 < x1 <
- Question : 11 - In this problem, we want to develop the two-dimensional spline interpolation procedure, which has applications in many areas such as image processing, weather maps, and topography analysis. Consider f (x, y) defined on [0, 4]

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