PadeApproximant

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`PadeApproximant`

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PadeApproximant[expr,{x,x₀,{m,n}}]

gives the Padé approximant to expr about the point x=x₀, with numerator order m and denominator order n.

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PadeApproximant[expr,{x,x₀,n}]

gives the diagonal Padé approximant to expr about the point x=x₀ of order n.

Details

The Wolfram Language can find the Padé approximant about the point x=x₀ only when it can evaluate power series at that point.
PadeApproximant produces a ratio of ordinary polynomial expressions, not a special SeriesData object.

Examples

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Basic Examples (2)Summary of the most common use cases

Order [2/3] Padé approximant for Exp[x]:

In[1]:=1

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https://wolfram.com/xid/0bh1avwzlboqs2-s6kt

Out[1]=1

PadeApproximant can handle functions with poles:

In[1]:=1

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https://wolfram.com/xid/0bh1avwzlboqs2-u6u0k

Out[1]=1

Scope (4)Survey of the scope of standard use cases

Padé approximant of an arbitrary function:

In[1]:=1

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https://wolfram.com/xid/0bh1avwzlboqs2-zr

Out[1]=1

Padé approximant with a complex-valued expansion point:

In[1]:=1

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https://wolfram.com/xid/0bh1avwzlboqs2-nqs

Out[1]=1

Padé approximant with an expansion point at infinity:

In[1]:=1

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https://wolfram.com/xid/0bh1avwzlboqs2-f3wj3b

Out[1]=1

Find a Padé approximant to a given series:

In[1]:=1

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https://wolfram.com/xid/0bh1avwzlboqs2-mgrang

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0bh1avwzlboqs2-lrujry

Out[2]=2

Generalizations & Extensions (3)Generalized and extended use cases

Padé approximant centered at the point :

In[1]:=1

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https://wolfram.com/xid/0bh1avwzlboqs2-cyy

Out[1]=1

Padé approximant in fractional powers:

In[1]:=1

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https://wolfram.com/xid/0bh1avwzlboqs2-h0w7al

Out[1]=1

Padé approximant of a function containing logarithmic terms:

In[1]:=1

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https://wolfram.com/xid/0bh1avwzlboqs2-kx7rd9

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0bh1avwzlboqs2-glpzij

Out[2]=2

Applications (2)Sample problems that can be solved with this function

Plot successive Padé approximants to :

In[1]:=1

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https://wolfram.com/xid/0bh1avwzlboqs2-ofv

Out[1]=1

Construct discrete orthogonal polynomials with respect to a discrete weighted measure:

In[1]:=1

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https://wolfram.com/xid/0bh1avwzlboqs2-brmvmh

In[2]:=2

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https://wolfram.com/xid/0bh1avwzlboqs2-hu3b2k

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0bh1avwzlboqs2-gbowa8

Plot the first few polynomials:

In[4]:=4

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https://wolfram.com/xid/0bh1avwzlboqs2-h4wgb4

Out[4]=4

Verify the orthogonality of the polynomials with respect to the measure:

In[5]:=5

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https://wolfram.com/xid/0bh1avwzlboqs2-ek3r4z

Out[5]=5

Properties & Relations (2)Properties of the function, and connections to other functions

The Padé approximant agrees with the ordinary series for terms:

In[1]:=1

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https://wolfram.com/xid/0bh1avwzlboqs2-x5l

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0bh1avwzlboqs2-x35

Out[2]=2

For PadeApproximant gives an ordinary series:

In[1]:=1

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https://wolfram.com/xid/0bh1avwzlboqs2-bv7

Out[1]=1

Possible Issues (2)Common pitfalls and unexpected behavior

Padé approximants often have spurious poles not present in the original function:

In[1]:=1

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https://wolfram.com/xid/0bh1avwzlboqs2-q8w

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0bh1avwzlboqs2-f7u

Out[2]=2

Padé approximants of a given order may not exist:

In[1]:=1

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https://wolfram.com/xid/0bh1avwzlboqs2-u53

Out[1]=1

Perturbing the order slightly is usually sufficient to produce an approximant:

In[2]:=2

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https://wolfram.com/xid/0bh1avwzlboqs2-um8

Out[2]=2

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