The file RationalApproximation.mth defines functions that compute Pade rational approximations and Chebyshev series approximations to an expression.  The function definitions in the file are automatically loaded when any of its functions are first used.


PADE(y, x, x0, n, d) simplifies to the Pade rational approximation to expression y depending on variable x, centered at x0, requesting a numerator of degree n and a denominator of degree d.  y must be differentiable through order n+d.


Pade approximations are often more efficient and have a larger region of convergence than Taylor series, particularly in the neighborhood of non-polynomial behavior in y, such as asymptotes.  For example,

PADE(#e^x, x, 0, 3, 3)

simplifies to the following cubic over cubic rational approximation of e^x centered on x=0:

    3       2               
   x  + 12·x  + 60·x + 120  
- ————————————————————————— 
    3       2               
   x  - 12·x  + 60·x - 120  


CHEBYSHEV_T_SERIES(u, x, a, b, k, n) simplifies to the Chebyshev series approximation of u(x) over the interval [a, b] up to x^k using the nth order Maclaurin series approximation for u(x).  The integer k must be less than or equal to n.  If k=n, CHEBYSHEV_T_SERIES simplifies to the nth order Maclaurin series approximation for u(x).  For example,

CHEBYSHEV_T_SERIES(EXP(x), x, -1/2, 1/2, 4, 5)

simplifies to the following 4th degree Chebyshev series approximation of e^x over the interval -1/2 to 1/2:

  4         3      2                 
 x      65·x      x      6143·x      
———— + ——————— + ———— + ———————— + 1 
 24      384       2      6144       


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