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numapprox[hermite_pade] - compute a Hermite-Pade approximation
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Calling Sequence
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hermite_pade([f1, f2,..., fn], x, N)
hermite_pade([f1, f2,..., fn], x, [d1, d2,..., dn])
hermite_pade([f1, f2,..., fn], x=a, N)
hermite_pade([f1, f2,..., fn], x=a, [d1, d2,..., dn])
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Parameters
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f1, ..., fn
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expressions representing the functions to be approximated
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x
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the variable appearing in the f_i's
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a
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the point about which to expand in a series
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N
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non-negative integer
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d1, ..., dn
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degree bounds
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Description
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The function hermite_pade computes a Hermite-Pade approximation of degree (d1,..., dn) for the functions f1,..., fn with respect to the variable x. When the degrees are not specified, but rather the order N is given, then an approximation of minimal degree is computed.
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Various levels of user information will be displayed during the computation if infolevel[hermite_pade] is assigned values between 1 and 3.
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This code is based on a procedure by H. Derksen in previous versions of the share library.
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The command with(numapprox,hermite_pade) allows the use of the abbreviated form of this command.
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Examples
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![hermite_pade([sin(x), cos(x), exp(x)], x = 0, [3, 2, 5])](/support/helpjp/helpview.aspx?si=7461/file04287/math183.png)
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![[-1275-255*x+45*x^2+5*x^3, 120+495*x+75*x^2, -120+900*x-600*x^2+160*x^3-20*x^4+x^5]](/support/helpjp/helpview.aspx?si=7461/file04287/math186.png)
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![hermite_pade([sin(x), cos(x)], x = Pi, 7)](/support/helpjp/helpview.aspx?si=7461/file04287/math190.png)
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![[15-6*(x-Pi)^2, -15*x+15*Pi+(x-Pi)^3]](/support/helpjp/helpview.aspx?si=7461/file04287/math193.png)
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![ff := [cos(2*x)*(x+1)+3, cos(x)^2+x*cos(x)+1, cos(2*x)+1, cos(x)]](/support/helpjp/helpview.aspx?si=7461/file04287/math197.png)
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![gg := [2, -4+2*x, -3*x, 4*x-2*x^2]](/support/helpjp/helpview.aspx?si=7461/file04287/math204.png)
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![simplify(ff[1]*gg[1]+ff[2]*gg[2]+ff[3]*gg[3]+ff[4]*gg[4])](/support/helpjp/helpview.aspx?si=7461/file04287/math208.png)
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| (4) |
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References
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Beckermann, B., and Labahn, G. "A uniform approach for Hermite Pade and simultaneous Pade approximants and their matrix-type generalizations." Numerical Algorithms, Vol. 3, (1992): 45-54.
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Beckermann, B., and Labahn, G. "A uniform approach for the fast computation of matrix-type Pade approximants." SIAM Journal on Matrix Analysis and Applications, Vol. 15, No. 3, (1994): 804-823.
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