8 Piecewise Polynomial Interpolation

Transcription

1 Applied Math Notes by R. J. LeVeque 8 Piecewise Polynomial Interpolation 8. Pitfalls of high order interpolation Suppose we know the value of a function at several points on an interval and we wish to find an interpolating function that we can use to approximate the function at all other points in the interval. We know from the previous section that if we have N + function values then we can find a unique polynomial of degree N that interpolates these values. We might hope that by adding more points we will get a better approximation, but this is not always the case, even if the function is very smooth. Figure 8. shows a famous example, a variant of an example first used by Runge about years ago to demonstrate the difficulty. The function f(x) = + x (8.) is shown in Figure 8.(a) over the interval 6 x 6. Figures (b,c,d) show the interpolating polynomials we obtain if we interpolate at N + equally spaced points in the interval [, ] for N + = 8,, 6. The interpolating polynomial looks good near the center of the interval, but not near the edges. Adding more points and using a higher degree polynomial only makes the problem worse, and the error in polynomial interpolation goes to infinity near the ends of the interval as N. There are two approaches to deriving a better interpolating polynomial. One approach is to use points that are not equally spaced. With a clever set of points one can get good high order polynomial approximations for this particular function and for other smooth functions. However, often we do not have any control over the interpolation points. A table of values or set of data is given to use at some fixed set of points and we need to use this to interpolate to other points. Another approach is to use some other form of interpolating polynomial rather than a high degree polynomial. In this section we will consider one other possibility, piecewise polynomials obtained by stitching together low order polynomials over subintervals of the interval of interest. 8. Piecewise Linear Interpolation We have already discussed piecewise linear interpolation in the context of the matlab plot command. Take another look at Figure 4.(b) and (c), which show piecewise linear functions approximating f(x) = sin(x) + cos(x). Now let s look at the error made in approximating a general function f(x) over some interval of interest [a,b] by a piecewise linear function interploting f at some N + points in the interval. In particular consider the equally spaced points x i = a + (i )h, for i =,,..., N +, where h = (b a)/n is the distance between grid points. These are the points that would be generated in matlab by the commands 8

2 Runge function Interpolating polynomial of degree (a) Interpolating polynomial of degree (b) Interpolating polynomial of degree (c) (d) Figure 8.: (a) The Runge function (8.). The other figures show three interpolating polynomials through (b) 8, (c), and (d) 6 equally spaced points. xdata = linspace(a,b,n+); or xdata = a:h:b; for example. Denote the piecewise linear function by p h (x), f[x ] + f[x,x ](x x ) if x x x, f[x ] + f[x,x ](x x ) if x x x, p h (x) = f[x ] + f[x,x 4 ](x x ) if x x x 4, f[x N ] + f[x N,x N+ ](x x N ) if x N x x N+. (8.) Now suppose we approximate f( x) for some x [a,b] by p h ( x), which is just the value obtained by linearly interpolating between the two closest values of x i where the function values of f are known. If x happens to be one of the points x i then the error is zero. Otherwise, suppose x j < x < x j+ for some index j. Then from (7.) we know that f( x) p h ( x) = f (η)( x x j )( x x j+ ). 8

3 for some η satisfying x j η x j+ and hence Moreover, we know that f( x) p h ( x) max x j z x j+ f (z) ( x x j )(x j+ x). ( x x j )(x j+ x) 4 h since this quadratic function takes its maximum when x =.(x j + x j+ ). Often we do not try to find the maximum of f over each subinterval. Instead we assume we have some bound for how large f is over the full interval [a,b]. Then we know that anywhere on the interval [a,b] the error will be bounded by or f( x) p h ( x) h 8 max a z b f (z), (8.) f( x) p h ( x) h 8 f for any x [a,b], (8.4) where for any function g (such as f ) we define the infinity norm of g on [a,b] by g = max g(z). (8.) a z b This is also often called the max norm. We ll see later why it s called the infinity norm. Since (8.4) holds for every x in the interval [a,b], we can also conclude that f p h h 8 f. (8.6) Note that for a given function f(x) and a fixed interval [a,b], the value of f is some fixed number that doesn t change if we change our interpolating function. In particular, it doesn t change if we do piecewise linear interpolation with more points. So (8.6) tells us how the error in this type of interpolation behaves as we take more and more points in the interval (i.e., as we make h smaller). The error goes to zero like some constant times h. So, for example, if we decrease h from. to. (i.e., use times as many points on the interval) then we expect the error to go down by a factor of. Example 8.. Recall that Figure 4. shows plots of piecewise linear approximations to the function (4.), f(x) = sin(x) + cos(x) over the interval [,] using points (h.), points (h /), and points (h ). For this function we can compute f (x) = sin(x) cos(x) and so an upper bound is f 7 since sin(x) and cos(x) are both bounded by. So we have an error bound.7 for h =, f p h.7h =.7 for h = /,.7 for h =.. With points or even points we expect a sizable error, as is seen in the plots of Figure 4.. For points (h.), on the other hand, the error is so small that it cannot possibly be observed with the resolution of the printer used to produce Figure 4. and for all practical purposes we have plotted the true function f(x). 8

4 8. Big-oh notation We say that f p h = O(h ) as h (8.7) using the big-oh notation that is often used in error analysis of this type. More generally, if h is a discretization parameter and E(h) is the error in a computed value resulting from a given value of h, we say that if there is some constant C, independent of h, so that E(h) = O(h p ) as h (8.8) E(h) Ch p (8.9) for all h > that are sufficiently small. Another way to say this is that E(h) h p C (8.) for all h sufficiently small, i.e., that E(h)/h p is uniformly bounded above as h. Note again that for a fixed function f the value f is independent of h, and so piecewise linear interpolation gives a max-norm error that is O(h ) for any function f C on the interval in question. The constant factor multiplying the h will depend on the particular function. We say that piecewise linear approximation is second order accurate since the error goes to zero like h for any f that is suitably smooth. By contrast piecewise quadratic approximation (discussed in the next section) is third order accurate; we will see that the error is O(h ) for any f that is sufficiently smooth (f C in this case). This type of analysis gives us a way to compare two different numerical algorithms (e.g. piecewise linear and piecewise quadratic interpolation) in general terms rather than just for a specific function f. 8.4 Piecewise Quadratic Interpolation As we have just seen, one way to obtain a more accurate interpolant is to increase the number of points and continue to use piecewise linear approximation. But we often don t have this luxury, for example when we have only a few tabulated values of the function and must interpolate from these. Another way to increase the accuracy is to increase the order of the polynomial approximations. Instead of a piecewise linear approximation we might try a piecewise quadratic interpolant. Determining a quadratic interpolant requires pieces of data. Suppose again that we have N + points where function values f(x i ) are known, and suppose N + is odd. Then there are an even number of subintervals, and we can pair these up into subintervals of length h on which values of f are known: at each end and in the middle. Our first subinterval is now [x,x ] and we know f(x ), f(x ), and f(x ). From these we can compute a quadratic polynomial and this polynomial will be used to approximate f(x) over [x,x ]. Similarly, on [x,x ] we compute a quadratic interpolating the values f(x ), f(x 4 ), and f(x ), and so on. The procedure yields a function p h (x) that is piecewise quadratic, given by a different quadratic polynomial on each interval [x j,x j+ ] for j =,,..., N/. To evaluate p h (x) 8 4

5 at any point we first determine what subinterval the point lies in and evaluate the appropriate quadratic function at the point. Here s the core of a matlab function pwquadinterp.m that takes as input a set of data points (xdata,ydata) and a (typically larger) set of points xi where we want to evaluate p h (x) and returns an array yi with the interpolated values: function yi = pwquadinterp(xdata,ydata,xi) yi = zeros(size(xi)); % initialize N = length(xdata)-; i = ; for j=::n+ % determine polynomial on interval [xdata(j-), xdata(j)] x = xdata(j-); x = xdata(j-); x = xdata(j); f = ydata(j-); f = ydata(j-); f = ydata(j); % compute divided differences: f = (f-f) / (x-x); f = (f-f) / (x-x); f = (f-f) / (x-x); % set function values for any xi(i) that lie in this interval: while i<=length(xi) & xi(i)<=x yi(i) = f + f*(xi(i)-x) + f*(xi(i)-x)*(xi(i)-x); i = i+; end end This routine has been cut down to the basics; a longer version on the webpage includes checks on the data to make sure an odd number of data points are provided and that the xdata and xi values are in increasing order, for example. Figure 8.(a) shows a plot of a simple data set and the piecewise linear interpolant p h (x), while Figure 8.(b) shows the piecewise quadratic interpolant for this same data, plotted at points by the following commands: xdata = :; ydata = [ ]; xi = linspace(,,); yi = pwquadinterp(xdata,ydata,xi); plot(xi,yi); hold on; plot(xdata,ydata, o ); hold off This data was purposely taken to be not very smooth so it would be clear where the different quadratic functions are joined together, at the points x =,, 7, 9,,. These points are called the knots of the piecewise polynomial. If smoother data were used, for example function values of some smooth function that is fairly well resolved at the data points, then the piecewise quadratic curve would look smoother. If f is a C function then error analysis similar to what we did for the piecewise linear case shows that, on a set of data points distance h apart, the error can be bounded by f p h Ch = O(h ) as h. The constant C depends on the size of the third derivative of f over the interval. Piecewise quadratic interpolation is rarely used in practice (though it forms the basis for Simpson s rule for quadrature, which we will study later). 8

6 8 piecewise linear 8 piecewise quadratic (a) (b) cubic spline 8 shape preserving piecewise cubic (c) (d) Figure 8.: (a) Piecewise linear function through data points. (b) Piecewise quadratic function through the same data points, with knots at x = ::. The function is defined by a single quadratic function on each interval indicated by dashed lines. (d) Cubic spline function through the same data points. This function is C (twice continuously differentiable). (c) Shape-preserving piecewise cubic function through the same data points. This function is C (the function and its first derivative are continuous. The second derivative is discontinuous at the knots x = :4). 8 6

7 8. Piecewise cubic interpolation A very popular form of piecewise polynomial interpolation is to use cubic functions on subintervals. A cubic function is determined by four pieces of data. However, rather than using four function values, a more common approach is to define a cubic function p i (x) on each subinterval [x i,x i+ ] by requiring the four conditions p i (x i ) = y i, p i (x i) = σ i, p i (x i+ ) = y i+, p i (x i+) = σ i+, (8.) where y i and y i+ are the desired function values at the ends of the interval (e.g., the function values f(x i ) and f(x i+ ) from a function we are interpolating) and σ i and σ i+ are the slopes of the function p i we wish to impose at the ends of the interval. If we know the derivative of f(x) then we might naturally use σ i = f (x i ) and σ i+ = f (x i+ ), but we will also consider other choices of these slopes below. Why do we choose to specify to function values and two slopes in this way? The main reason is that the resulting piecewise cubic function will not only be continuous at the knots x i, since p,i (x i ) = p i (x i ) = y i, (8.) but also the derivative of the piecewise cubic function will be continuous, p,i (x i ) = p i(x i ) = σ i, (8.) since the same slope condition is imposed on both cubics that meet at x i. A piecewise cubic approximation of this form will thus look smoother than a piecewise quadratic approximation of the form discussed above. For the moment suppose we know the four data values on the right hand side of (8.). How would we determine p i (x)? There are various ways. For example we could write p i (x) = c + c x + c x + c x and evaluate this function and its derivative p i (x) = c + c x + c x at the two endpoints of the subinterval. Requiring that (8.) be satisfied would then give four equations in the four unknowns c, c, c, and c. An easier way is to again use a divided difference table and the Newton form. This requires some modification to incorporate the slope data. We compute the following table: x i f[x i ] x i f[x i ] f[x i,x i ] x i+ f[x i+ ] f[x i,x i+ ] f[x i,x i,x i+ ] x i+ f[x i+ ] f[x i+,x i+ ] f[x i,x i+,x i+ ] f[x i,x i,x i+,x i+ ] (8.4) Note that each point is repeated twice since we have two pieces of information. The divided differences are computed using the usual formula (7.6) with two exceptions: we use f[x i,x i ] = σ i and f[x i+,x i+ ] = σ i+. Note that these values are not defined properly by the usual formula f[x,x ] = (f(x ) f(x ))/(x x ) since this is /. However, in the limit as x x this approaches f (x ), so it makes sense to replace this first order divided difference by the derivative value in this case. 8 7

8 The other values in (8.4) are computed by (7.6), e.g., f[x i,x i,x i+ ] = f[x i,x i+ ] f[x i,x i ] x i+ x i = The interpolating polynomial also has the usual form, f(x i+ ) f(x i ) x i+ x i σ i. x i+ x i p i (x) = f[x i ] + f[x i,x i ](x x i ) + f[x i,x i,x i+ ](x x i ) + f[x i,x i,x i+,x i+ ](x x i ) (x x i+ ). (8.) This is sometimes called Hermite cubic interpolation. How do we choose the values σ i if we don t know the exact derivative f (x i )? One way is to base σ i on approximations to f (x i ) obtained using divided differences. Another very popular method is to use a cubic spline. With this method the values σ i are chosen by requiring that the resulting piecewise cubic function have continuous second derivatives at the knots as well as continuity of the function and first derivative. This gives a very smooth looking function. It turns out it is always possible to choose σ i values so that the second derivative is continuous, but doing so requires solving a linear system of equations for all the σ i values based on all the data. We will not investigate this method in more detail, but it is often used in practice and can be computed using the matlab function described in the next section. 8.6 Piecewise polynomials in matlab In matlab there is a function interp that interpolates data in dimension by piecewise polynomials. It can be called in the same manner as the function pwquadinterp defined above. It takes another argument, a string that specifies the method to use. The command y = interp(xdata,ydata,x, linear ); uses piecewise linear interpolation based on the data xdata, ydata and evaluates the resulting interpolating function at all points in x. The command y = interp(xdata,ydata,x, spline ); determines and evaluates the cubic spline. There is another option, y = interp(xdata,ydata,x, cubic ); which uses shape preserving cubic interpolation, which chooses the slopes σ i in a way that avoids introducing oscillations in the function that are not in the data. Figure 8.(c) and (d) show the two piecewise cubics for one set of data. There is another form of interp that may also be useful: p = interp(xdata,ydata, cubic, pp ); for example, determines the data needed to construct the cubic spline p and returns a data structure with all this information, but doesn t evaluate this function at any points. Once p has been computed, it can be evaluated anywhere desired using y = ppval(p, x); 8 8

9 piecewise linear cubic spline Figure 8.: (a) Piecewise linear function through data points on a spiral curve. (b) Cubic spline function through the same data points. 8.7 Parameterized curves So far we have always used the plot command to plot graphs of functions, but plot can be used to plot any curve in the plane. Often we want to plot a smooth curve but only know a few points on the curve and would like to interpolate between them. We can do this by viewing the points (x,y) on the curve as being functions of some parameter t, so the curve is defined by (x(t),y(t)). We can then view the given data points as being values of these functions at a few points, and interpolate in t to obtain values in between. Here s an example: tdata = linspace(,6*pi,); xdata = tdata.*cos(tdata); ydata = tdata.*sin(tdata); plot(xdata,ydata, ro ); % plot the points t = linspace (, 6*pi, ); x = interp(tdata,xdata,t, spline ); y = interp(tdata,ydata,t, spline ); hold on; plot(x,y) % plot the smooth curve axis([- - ]); daspect([ ]) This defines points on a spiral curve and then evaluates the cubic spline approximations to x(t) and y(t) at points to produce the smooth curve shown in Figure 8.(b). If we simply performed plot(xdata,ydata) we would instead get Figure 8.(a). 8 9