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The Maxima on-line user's manual

Algebra Calculator

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Pade Calculator

Pade

Function: pade (<taylor_series>, <numer_deg_bound>, <denom_deg_bound>) Returns a list of all rational functions which have the given Taylor series expansion where the sum of the degrees of the numerator and the denominator is less than or equal to the truncation level of the power series, i.e. are "best" approximants, and which additionally satisfy the specified degree bounds.

eval(pade(taylor(sin(x),x,0,19),19,1));

<taylor_series> is a univariate Taylor series. <numer_deg_bound> and <denom_deg_bound> are positive integers specifying degree bounds on the numerator and denominator.

<taylor_series> can also be a Laurent series, and the degree bounds can be inf which causes all rational functions whose total degree is less than or equal to the length of the power series to be returned. Total degree is defined as <numer_deg_bound> + <denom_deg_bound>. Length of a power series is defined as "truncation level" + 1 - min(0, "order of series").

          (%i1) taylor (1 + x + x^2 + x^3, x, 0, 3);
                                        2    3
          (%o1)/T/             1 + x + x  + x  + . . .
          (%i2) pade (%, 1, 1);
                                           1
          (%o2)                       [- -----]
                                         x - 1
          (%i3) t: taylor(-(83787*x^10 - 45552*x^9 - 187296*x^8
                             + 387072*x^7 + 86016*x^6 - 1507328*x^5
                             + 1966080*x^4 + 4194304*x^3 - 25165824*x^2
                             + 67108864*x - 134217728)
                 /134217728, x, 0, 10);
                              2    3       4       5       6        7
                       x   3 x    x    15 x    23 x    21 x    189 x
          (%o3)/T/ 1 - - + ---- - -- - ----- + ----- - ----- - ------
                       2    16    32   1024    2048    32768   65536

8 9 10 5853 x 2847 x 83787 x + ------- + ------- - --------- + . . . 4194304 8388608 134217728

          (%i4) pade (t, 4, 4);
          (%o4)                          []

There is no rational function of degree 4 numerator/denominator, with this power series expansion. You must in general have degree of the numerator and degree of the denominator adding up to at least the degree of the power series, in order to have enough unknown coefficients to solve.

          (%i5) pade (t, 5, 5);
                               5                4                 3
          (%o5) [- (520256329 x  - 96719020632 x  - 489651410240 x

2 - 1619100813312 x - 2176885157888 x - 2386516803584)

5 4 3 /(47041365435 x + 381702613848 x + 1360678489152 x

2 + 2856700692480 x + 3370143559680 x + 2386516803584)]

(%o1)                                true
(%i2) 

Pade Example

Related Examples

pade-tanh-taylor

DISPLAY2D = False;

pade (taylor (tanh (x...

Calculate

pade-plot2d-taylor
plot2d([f(x), p(x)], [x, -1, 1]);

f(x) := %e^(-x*x);

n : 28;

g(x) := taylor(f(x), ...

Calculate

pade-tanh-taylor

t: taylor(tanh(x), x,...

pade (t, 3, 4);

Calculate

pade-plot2d-taylor
plot2d([f(x)/p(x)-1], [x, -1, 1]);

f(x) := %e^(-x*x);

n : 32;

g(x) := taylor(f(x), ...

Calculate

pade-taylor

pade(taylor(arctan(x*...

Calculate

pade-tan-taylor

h:taylor(tan(x),x,1.5...

pade(h,5,5);

Calculate

pade-plot2d-taylor
plot2d([f(x), p(x)], [x, -1, 1]);

f(x) := %e^(-x*x);

n : 28;

g(x) := taylor(f(x), ...

Calculate

pade-taylor

t: taylor ( (-x^2-x)/...

pade (t, 2, 2);

Calculate

pade-taylor

f(x) := e^(-x*x);

n : 64;

g(x) := taylor(f(x), ...

Calculate

pade-plot2d-sin-taylor
plot2d([f(x),h,d[1]],[x,-5,5],[y,-5,15]);

f(x):=sin(x);

h:taylor(f(x),x,0,10);

d:pade(h,5,5);

Calculate