Derivatives or Integrals of a Chebyshev-approximated Function If you have obtained the Chebyshev coefficients that approximate a function in a certain range (., from chebft in §), then it is a simple matter to transform them to Chebyshev coefficients corresponding to the derivative or integral of the function. Having done this, you can evaluate the derivative or integral just as if it were a function that you had Chebyshev-fitted ab initio. The relevant formulas are these: If ci , i = 0, . . . , m − 1 are the coefficients that approximate a function f in equation (). | Derivatives orIntegrals of a Chebyshev-approximatedFunction 195 Derivatives or Integrals of a Chebyshev-approximatedFunction If you have obtained the Chebyshev coefficients that approximate a function in a certain range . from chebft in then it is a simple matter to transform them to Chebyshev coefficients corresponding to the derivative or integral of the function. Having done this you can evaluate the derivative or integral just as if it were a function that you had Chebyshev-fitted ab initio. The relevant formulas are these If ci i 0 . . m 1 are the coefficients that approximate a function f in equation Ci are the coefficients that approximate the indefinite integral of f and ci are the coefficients that approximate the derivative of f then Ci-1 Ci 1 Ci 2 i 1 i 11 5-S 1 ci_1 C0 1 2 i 1 ci i m 1 m 2 . . . 2 Equation is augmented by an arbitrary choice of Co corresponding to an arbitrary constant of integration. Equation which is a recurrence is started with the values c m c m-1 0 corresponding to no information about the m 1st Chebyshev coefficient of the original function f. Here are routines for implementing equations and . void chder float a float b float c float cder int n Given a b c as output from routine chebft and given n the desired degree of approximation length of c to be used this routine returns the array cder the Chebyshev coefficients of the derivative of the function whose coefficients are c. int j float con cder n-1 n-1 and n-2 are special cases. cder n-2 2 n-1 c n-1 for j n-3 j 0 j-- cder j cder j 2 2 j 1 c j 1 Equation . con b-a for j 0 j n j Normalize to the interval b-a. cder j con void chint float a float b float c float cint int n Given a b c as output from routine chebft and given n the desired degree of approximation length of c to be used this routine returns the array cint the Chebyshev coefficients of the integral of the function whose .
