import numpy as np from scipy import factorial __all__ = ["KroghInterpolator", "krogh_interpolate", "BarycentricInterpolator", "barycentric_interpolate", "PiecewisePolynomial", "piecewise_polynomial_interpolate","approximate_taylor_polynomial"] class KroghInterpolator(object): """The interpolating polynomial for a set of points Constructs a polynomial that passes through a given set of points, optionally with specified derivatives at those points. Allows evaluation of the polynomial and all its derivatives. For reasons of numerical stability, this function does not compute the coefficients of the polynomial, although they can be obtained by evaluating all the derivatives. Be aware that the algorithms implemented here are not necessarily the most numerically stable known. Moreover, even in a world of exact computation, unless the x coordinates are chosen very carefully - Chebyshev zeros (e.g. cos(i*pi/n)) are a good choice - polynomial interpolation itself is a very ill-conditioned process due to the Runge phenomenon. In general, even with well-chosen x values, degrees higher than about thirty cause problems with numerical instability in this code. Based on Krogh 1970, "Efficient Algorithms for Polynomial Interpolation and Numerical Differentiation" """ def __init__(self, xi, yi): """Construct an interpolator passing through the specified points The polynomial passes through all the pairs (xi,yi). One may additionally specify a number of derivatives at each point xi; this is done by repeating the value xi and specifying the derivatives as successive yi values. Parameters ---------- xi : array-like, length N known x-coordinates yi : array-like, N by R known y-coordinates, interpreted as vectors of length R, or scalars if R=1 Example ------- To produce a polynomial that is zero at 0 and 1 and has derivative 2 at 0, call >>> KroghInterpolator([0,0,1],[0,2,0]) """ self.xi = np.asarray(xi) self.yi = np.asarray(yi) if len(self.yi.shape)==1: self.vector_valued = False self.yi = self.yi[:,np.newaxis] elif len(self.yi.shape)>2: raise ValueError("y coordinates must be either scalars or vectors") else: self.vector_valued = True n = len(xi) self.n = n nn, r = self.yi.shape if nn!=n: raise ValueError("%d x values provided and %d y values; must be equal" % (n, nn)) self.r = r c = np.zeros((n+1,r)) c[0] = yi[0] Vk = np.zeros((n,r)) for k in xrange(1,n): s = 0 while s<=k and xi[k-s]==xi[k]: s += 1 s -= 1 Vk[0] = yi[k]/float(factorial(s)) for i in xrange(k-s): assert xi[i]!=xi[k] if s==0: Vk[i+1] = (c[i]-Vk[i])/(xi[i]-xi[k]) else: Vk[i+1] = (Vk[i+1]-Vk[i])/(xi[i]-xi[k]) c[k] = Vk[k-s] self.c = c def __call__(self,x): """Evaluate the polynomial at the point x Parameters ---------- x : scalar or array-like of length N Returns ------- y : scalar, array of length R, array of length N, or array of length N by R If x is a scalar, returns either a vector or a scalar depending on whether the interpolator is vector-valued or scalar-valued. If x is a vector, returns a vector of values. """ if np.isscalar(x): scalar = True m = 1 else: scalar = False m = len(x) x = np.asarray(x) n = self.n pi = 1 p = np.zeros((m,self.r)) p += self.c[0,np.newaxis,:] for k in xrange(1,n): w = x - self.xi[k-1] pi = w*pi p = p + np.multiply.outer(pi,self.c[k]) if not self.vector_valued: if scalar: return p[0,0] else: return p[:,0] else: if scalar: return p[0] else: return p def derivatives(self,x,der=None): """Evaluate many derivatives of the polynomial at the point x Produce an array of all derivative values at the point x. Parameters ---------- x : scalar or array-like of length N Point or points at which to evaluate the derivatives der : None or integer How many derivatives to extract; None for all potentially nonzero derivatives (that is a number equal to the number of points). This number includes the function value as 0th derivative. Returns ------- d : array If the interpolator's values are R-dimensional then the returned array will be der by N by R. If x is a scalar, the middle dimension will be dropped; if R is 1 then the last dimension will be dropped. Example ------- >>> KroghInterpolator([0,0,0],[1,2,3]).derivatives(0) array([1.0,2.0,3.0]) >>> KroghInterpolator([0,0,0],[1,2,3]).derivatives([0,0]) array([[1.0,1.0], [2.0,2.0], [3.0,3.0]]) """ if np.isscalar(x): scalar = True m = 1 else: scalar = False m = len(x) x = np.asarray(x) n = self.n r = self.r if der is None: der = self.n dern = min(self.n,der) pi = np.zeros((n,m)) w = np.zeros((n,m)) pi[0] = 1 p = np.zeros((m,self.r)) p += self.c[0,np.newaxis,:] for k in xrange(1,n): w[k-1] = x - self.xi[k-1] pi[k] = w[k-1]*pi[k-1] p += np.multiply.outer(pi[k],self.c[k]) cn = np.zeros((max(der,n+1),m,r)) cn[:n+1,...] += self.c[:n+1,np.newaxis,:] cn[0] = p for k in xrange(1,n): for i in xrange(1,n-k+1): pi[i] = w[k+i-1]*pi[i-1]+pi[i] cn[k] = cn[k]+pi[i,:,np.newaxis]*cn[k+i] cn[k]*=factorial(k) cn[n,...] = 0 if not self.vector_valued: if scalar: return cn[:der,0,0] else: return cn[:der,:,0] else: if scalar: return cn[:der,0] else: return cn[:der] def derivative(self,x,der): """Evaluate one derivative of the polynomial at the point x Parameters ---------- x : scalar or array-like of length N Point or points at which to evaluate the derivatives der : None or integer Which derivative to extract. This number includes the function value as 0th derivative. Returns ------- d : array If the interpolator's values are R-dimensional then the returned array will be N by R. If x is a scalar, the middle dimension will be dropped; if R is 1 then the last dimension will be dropped. Notes ----- This is computed by evaluating all derivatives up to the desired one (using self.derivatives()) and then discarding the rest. """ return self.derivatives(x,der=der+1)[der] def krogh_interpolate(xi,yi,x,der=0): """Convenience function for polynomial interpolation. Constructs a polynomial that passes through a given set of points, optionally with specified derivatives at those points. Evaluates the polynomial or some of its derivatives. For reasons of numerical stability, this function does not compute the coefficients of the polynomial, although they can be obtained by evaluating all the derivatives. Be aware that the algorithms implemented here are not necessarily the most numerically stable known. Moreover, even in a world of exact computation, unless the x coordinates are chosen very carefully - Chebyshev zeros (e.g. cos(i*pi/n)) are a good choice - polynomial interpolation itself is a very ill-conditioned process due to the Runge phenomenon. In general, even with well-chosen x values, degrees higher than about thirty cause problems with numerical instability in this code. Based on Krogh 1970, "Efficient Algorithms for Polynomial Interpolation and Numerical Differentiation" The polynomial passes through all the pairs (xi,yi). One may additionally specify a number of derivatives at each point xi; this is done by repeating the value xi and specifying the derivatives as successive yi values. Parameters ---------- xi : array-like, length N known x-coordinates yi : array-like, N by R known y-coordinates, interpreted as vectors of length R, or scalars if R=1 x : scalar or array-like of length N Point or points at which to evaluate the derivatives der : integer or list How many derivatives to extract; None for all potentially nonzero derivatives (that is a number equal to the number of points), or a list of derivatives to extract. This number includes the function value as 0th derivative. Returns ------- d : array If the interpolator's values are R-dimensional then the returned array will be the number of derivatives by N by R. If x is a scalar, the middle dimension will be dropped; if the yi are scalars then the last dimension will be dropped. Notes ----- Construction of the interpolating polynomial is a relatively expensive process. If you want to evaluate it repeatedly consider using the class KroghInterpolator (which is what this function uses). """ P = KroghInterpolator(xi, yi) if der==0: return P(x) elif np.isscalar(der): return P.derivative(x,der=der) else: return P.derivatives(x,der=np.amax(der)+1)[der] def approximate_taylor_polynomial(f,x,degree,scale,order=None): """Estimate the Taylor polynomial of f at x by polynomial fitting A polynomial Parameters ---------- f : callable The function whose Taylor polynomial is sought. Should accept a vector of x values. x : scalar The point at which the polynomial is to be evaluated. degree : integer The degree of the Taylor polynomial scale : scalar The width of the interval to use to evaluate the Taylor polynomial. Function values spread over a range this wide are used to fit the polynomial. Must be chosen carefully. order : integer or None The order of the polynomial to be used in the fitting; f will be evaluated order+1 times. If None, use degree. Returns ------- p : poly1d the Taylor polynomial (translated to the origin, so that for example p(0)=f(x)). Notes ----- The appropriate choice of "scale" is a tradeoff - too large and the function differs from its Taylor polynomial too much to get a good answer, too small and roundoff errors overwhelm the higher-order terms. The algorithm used becomes numerically unstable around order 30 even under ideal circumstances. Choosing order somewhat larger than degree may improve the higher-order terms. """ if order is None: order=degree n = order+1 # Choose n points that cluster near the endpoints of the interval in # a way that avoids the Runge phenomenon. Ensure, by including the # endpoint or not as appropriate, that one point always falls at x # exactly. xs = scale*np.cos(np.linspace(0,np.pi,n,endpoint=n%1)) + x P = KroghInterpolator(xs, f(xs)) d = P.derivatives(x,der=degree+1) return np.poly1d((d/factorial(np.arange(degree+1)))[::-1]) class BarycentricInterpolator(object): """The interpolating polynomial for a set of points Constructs a polynomial that passes through a given set of points. Allows evaluation of the polynomial, efficient changing of the y values to be interpolated, and updating by adding more x values. For reasons of numerical stability, this function does not compute the coefficients of the polynomial. This class uses a "barycentric interpolation" method that treats the problem as a special case of rational function interpolation. This algorithm is quite stable, numerically, but even in a world of exact computation, unless the x coordinates are chosen very carefully - Chebyshev zeros (e.g. cos(i*pi/n)) are a good choice - polynomial interpolation itself is a very ill-conditioned process due to the Runge phenomenon. Based on Berrut and Trefethen 2004, "Barycentric Lagrange Interpolation". """ def __init__(self, xi, yi=None): """Construct an object capable of interpolating functions sampled at xi The values yi need to be provided before the function is evaluated, but none of the preprocessing depends on them, so rapid updates are possible. Parameters ---------- xi : array-like of length N The x coordinates of the points the polynomial should pass through yi : array-like N by R or None The y coordinates of the points the polynomial should pass through; if R>1 the polynomial is vector-valued. If None the y values will be supplied later. """ self.n = len(xi) self.xi = np.asarray(xi) if yi is not None and len(yi)!=len(self.xi): raise ValueError("yi dimensions do not match xi dimensions") self.set_yi(yi) self.wi = np.zeros(self.n) self.wi[0] = 1 for j in xrange(1,self.n): self.wi[:j]*=(self.xi[j]-self.xi[:j]) self.wi[j] = np.multiply.reduce(self.xi[:j]-self.xi[j]) self.wi**=-1 def set_yi(self, yi): """Update the y values to be interpolated The barycentric interpolation algorithm requires the calculation of weights, but these depend only on the xi. The yi can be changed at any time. Parameters ---------- yi : array-like N by R The y coordinates of the points the polynomial should pass through; if R>1 the polynomial is vector-valued. If None the y values will be supplied later. """ if yi is None: self.yi = None return yi = np.asarray(yi) if len(yi.shape)==1: self.vector_valued = False yi = yi[:,np.newaxis] elif len(yi.shape)>2: raise ValueError("y coordinates must be either scalars or vectors") else: self.vector_valued = True n, r = yi.shape if n!=len(self.xi): raise ValueError("yi dimensions do not match xi dimensions") self.yi = yi self.r = r def add_xi(self, xi, yi=None): """Add more x values to the set to be interpolated The barycentric interpolation algorithm allows easy updating by adding more points for the polynomial to pass through. Parameters ---------- xi : array-like of length N1 The x coordinates of the points the polynomial should pass through yi : array-like N1 by R or None The y coordinates of the points the polynomial should pass through; if R>1 the polynomial is vector-valued. If None the y values will be supplied later. The yi should be specified if and only if the interpolator has y values specified. """ if yi is not None: if self.yi is None: raise ValueError("No previous yi value to update!") yi = np.asarray(yi) if len(yi.shape)==1: if self.vector_valued: raise ValueError("Cannot extend dimension %d y vectors with scalars" % self.r) yi = yi[:,np.newaxis] elif len(yi.shape)>2: raise ValueError("y coordinates must be either scalars or vectors") else: n, r = yi.shape if r!=self.r: raise ValueError("Cannot extend dimension %d y vectors with dimension %d y vectors" % (self.r, r)) self.yi = np.vstack((self.yi,yi)) else: if self.yi is not None: raise ValueError("No update to yi provided!") old_n = self.n self.xi = np.concatenate((self.xi,xi)) self.n = len(self.xi) self.wi**=-1 old_wi = self.wi self.wi = np.zeros(self.n) self.wi[:old_n] = old_wi for j in xrange(old_n,self.n): self.wi[:j]*=(self.xi[j]-self.xi[:j]) self.wi[j] = np.multiply.reduce(self.xi[:j]-self.xi[j]) self.wi**=-1 def __call__(self, x): """Evaluate the interpolating polynomial at the points x Parameters ---------- x : scalar or array-like of length M Returns ------- y : scalar or array-like of length R or length M or M by R The shape of y depends on the shape of x and whether the interpolator is vector-valued or scalar-valued. Notes ----- Currently the code computes an outer product between x and the weights, that is, it constructs an intermediate array of size N by M, where N is the degree of the polynomial. """ scalar = np.isscalar(x) x = np.atleast_1d(x) c = np.subtract.outer(x,self.xi) z = c==0 c[z] = 1 c = self.wi/c p = np.dot(c,self.yi)/np.sum(c,axis=-1)[:,np.newaxis] i, j = np.nonzero(z) p[i] = self.yi[j] if not self.vector_valued: if scalar: return p[0,0] else: return p[:,0] else: if scalar: return p[0] else: return p def barycentric_interpolate(xi, yi, x): """Convenience function for polynomial interpolation Constructs a polynomial that passes through a given set of points, then evaluates the polynomial. For reasons of numerical stability, this function does not compute the coefficients of the polynomial. This function uses a "barycentric interpolation" method that treats the problem as a special case of rational function interpolation. This algorithm is quite stable, numerically, but even in a world of exact computation, unless the x coordinates are chosen very carefully - Chebyshev zeros (e.g. cos(i*pi/n)) are a good choice - polynomial interpolation itself is a very ill-conditioned process due to the Runge phenomenon. Based on Berrut and Trefethen 2004, "Barycentric Lagrange Interpolation". Parameters ---------- xi : array-like of length N The x coordinates of the points the polynomial should pass through yi : array-like N by R The y coordinates of the points the polynomial should pass through; if R>1 the polynomial is vector-valued. x : scalar or array-like of length M Returns ------- y : scalar or array-like of length R or length M or M by R The shape of y depends on the shape of x and whether the interpolator is vector-valued or scalar-valued. Notes ----- Construction of the interpolation weights is a relatively slow process. If you want to call this many times with the same xi (but possibly varying yi or x) you should use the class BarycentricInterpolator. This is what this function uses internally. """ return BarycentricInterpolator(xi, yi)(x) class PiecewisePolynomial(object): """Piecewise polynomial curve specified by points and derivatives This class represents a curve that is a piecewise polynomial. It passes through a list of points and has specified derivatives at each point. The degree of the polynomial may very from segment to segment, as may the number of derivatives available. The degree should not exceed about thirty. Appending points to the end of the curve is efficient. """ def __init__(self, xi, yi, orders=None, direction=None): """Construct a piecewise polynomial Parameters ---------- xi : array-like of length N a sorted list of x-coordinates yi : list of lists of length N yi[i] is the list of derivatives known at xi[i] orders : list of integers, or integer a list of polynomial orders, or a single universal order direction : {None, 1, -1} indicates whether the xi are increasing or decreasing +1 indicates increasing -1 indicates decreasing None indicates that it should be deduced from the first two xi Notes ----- If orders is None, or orders[i] is None, then the degree of the polynomial segment is exactly the degree required to match all i available derivatives at both endpoints. If orders[i] is not None, then some derivatives will be ignored. The code will try to use an equal number of derivatives from each end; if the total number of derivatives needed is odd, it will prefer the rightmost endpoint. If not enough derivatives are available, an exception is raised. """ yi0 = np.asarray(yi[0]) if len(yi0.shape)==2: self.vector_valued = True self.r = yi0.shape[1] elif len(yi0.shape)==1: self.vector_valued = False self.r = 1 else: raise ValueError("Each derivative must be a vector, not a higher-rank array") self.xi = [xi[0]] self.yi = [yi0] self.n = 1 self.direction = direction self.orders = [] self.polynomials = [] self.extend(xi[1:],yi[1:],orders) def _make_polynomial(self,x1,y1,x2,y2,order,direction): """Construct the interpolating polynomial object Deduces the number of derivatives to match at each end from order and the number of derivatives available. If possible it uses the same number of derivatives from each end; if the number is odd it tries to take the extra one from y2. In any case if not enough derivatives are available at one end or another it draws enough to make up the total from the other end. """ n = order+1 n1 = min(n//2,len(y1)) n2 = min(n-n1,len(y2)) n1 = min(n-n2,len(y1)) if n1+n2!=n: raise ValueError("Point %g has %d derivatives, point %g has %d derivatives, but order %d requested" % (x1, len(y1), x2, len(y2), order)) assert n1<=len(y1) assert n2<=len(y2) xi = np.zeros(n) if self.vector_valued: yi = np.zeros((n,self.r)) else: yi = np.zeros((n,)) xi[:n1] = x1 yi[:n1] = y1[:n1] xi[n1:] = x2 yi[n1:] = y2[:n2] return KroghInterpolator(xi,yi) def append(self, xi, yi, order=None): """Append a single point with derivatives to the PiecewisePolynomial Parameters ---------- xi : float yi : array-like yi is the list of derivatives known at xi order : integer or None a polynomial order, or instructions to use the highest possible order """ yi = np.asarray(yi) if self.vector_valued: if (len(yi.shape)!=2 or yi.shape[1]!=self.r): raise ValueError("Each derivative must be a vector of length %d" % self.r) else: if len(yi.shape)!=1: raise ValueError("Each derivative must be a scalar") if self.direction is None: self.direction = np.sign(xi-self.xi[-1]) elif (xi-self.xi[-1])*self.direction < 0: raise ValueError("x coordinates must be in the %d direction: %s" % (self.direction, self.xi)) self.xi.append(xi) self.yi.append(yi) if order is None: n1 = len(self.yi[-2]) n2 = len(self.yi[-1]) n = n1+n2 order = n-1 self.orders.append(order) self.polynomials.append(self._make_polynomial( self.xi[-2], self.yi[-2], self.xi[-1], self.yi[-1], order, self.direction)) self.n += 1 def extend(self, xi, yi, orders=None): """Extend the PiecewisePolynomial by a list of points Parameters ---------- xi : array-like of length N1 a sorted list of x-coordinates yi : list of lists of length N1 yi[i] is the list of derivatives known at xi[i] orders : list of integers, or integer a list of polynomial orders, or a single universal order direction : {None, 1, -1} indicates whether the xi are increasing or decreasing +1 indicates increasing -1 indicates decreasing None indicates that it should be deduced from the first two xi """ for i in xrange(len(xi)): if orders is None or np.isscalar(orders): self.append(xi[i],yi[i],orders) else: self.append(xi[i],yi[i],orders[i]) def __call__(self, x): """Evaluate the piecewise polynomial Parameters ---------- x : scalar or array-like of length N Returns ------- y : scalar or array-like of length R or length N or N by R """ if np.isscalar(x): pos = np.clip(np.searchsorted(self.xi, x) - 1, 0, self.n-2) y = self.polynomials[pos](x) else: x = np.asarray(x) m = len(x) pos = np.clip(np.searchsorted(self.xi, x) - 1, 0, self.n-2) if self.vector_valued: y = np.zeros((m,self.r)) else: y = np.zeros(m) for i in xrange(self.n-1): c = pos==i y[c] = self.polynomials[i](x[c]) return y def derivative(self, x, der): """Evaluate a derivative of the piecewise polynomial Parameters ---------- x : scalar or array-like of length N der : integer which single derivative to extract Returns ------- y : scalar or array-like of length R or length N or N by R Notes ----- This currently computes (using self.derivatives()) all derivatives of the curve segment containing each x but returns only one. """ return self.derivatives(x,der=der+1)[der] def derivatives(self, x, der): """Evaluate a derivative of the piecewise polynomial Parameters ---------- x : scalar or array-like of length N der : integer how many derivatives (including the function value as 0th derivative) to extract Returns ------- y : array-like of shape der by R or der by N or der by N by R """ if np.isscalar(x): pos = np.clip(np.searchsorted(self.xi, x) - 1, 0, self.n-2) y = self.polynomials[pos].derivatives(x,der=der) else: x = np.asarray(x) m = len(x) pos = np.clip(np.searchsorted(self.xi, x) - 1, 0, self.n-2) if self.vector_valued: y = np.zeros((der,m,self.r)) else: y = np.zeros((der,m)) for i in xrange(self.n-1): c = pos==i y[:,c] = self.polynomials[i].derivatives(x[c],der=der) return y def piecewise_polynomial_interpolate(xi,yi,x,orders=None,der=0): """Convenience function for piecewise polynomial interpolation Parameters ---------- xi : array-like of length N a sorted list of x-coordinates yi : list of lists of length N yi[i] is the list of derivatives known at xi[i] x : scalar or array-like of length M orders : list of integers, or integer a list of polynomial orders, or a single universal order der : integer which single derivative to extract Returns ------- y : scalar or array-like of length R or length M or M by R Notes ----- If orders is None, or orders[i] is None, then the degree of the polynomial segment is exactly the degree required to match all i available derivatives at both endpoints. If orders[i] is not None, then some derivatives will be ignored. The code will try to use an equal number of derivatives from each end; if the total number of derivatives needed is odd, it will prefer the rightmost endpoint. If not enough derivatives are available, an exception is raised. Construction of these piecewise polynomials can be an expensive process; if you repeatedly evaluate the same polynomial, consider using the class PiecewisePolynomial (which is what this function does). """ P = PiecewisePolynomial(xi, yi, orders) if der==0: return P(x) elif np.isscalar(der): return P.derivative(x,der=der) else: return P.derivatives(x,der=np.amax(der)+1)[der]