Padé approximant - Revision history

Vipul at 00:57, 2 May 2014

2014-05-02T00:57:13Z

← Older revision		Revision as of 00:57, 2 May 2014
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	where <math>a_0,a_1,\dots,a_m,b_1,b_2,\dots,b_n</math> are all real numbers, and where <math>f^{(j)}(0) = R^{(j)}(0)</math> for <math>j \in \{ 0, 1,2,\dots,m+n\}</math>.		where <math>a_0,a_1,\dots,a_m,b_1,b_2,\dots,b_n</math> are all real numbers, and where <math>f^{(j)}(0) = R^{(j)}(0)</math> for <math>j \in \{ 0, 1,2,\dots,m+n\}</math>.

			{{under construction check wikipedia}}

Vipul at 00:56, 2 May 2014

2014-05-02T00:56:27Z

← Older revision		Revision as of 00:56, 2 May 2014
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	===About a general point and for a given order===		===About a general point and for a given order===

	Suppose <math>f</math> is a function, <math>x_0</math> is a point in the domain of <math>f</math>, and <math>m,n</math> are (possibly equal, possibly distinct) nonnegative integers. The '''Padé approximant''' to <math>f</math> of order <math>[m/n]</math> at <math>x_0</math> is a rational function of the form:		Suppose <math>f</math> is a function, <math>x_0</math> is a point in the domain of <math>f</math>, and <math>m,n</math> are (possibly equal, possibly distinct) nonnegative integers. Suppose further that <math>f</math> is at least <math>m + n</math> times differentiable at <math>x_0</math>.

			The '''Padé approximant''' to <math>f</math> of order <math>[m/n]</math> at <math>x_0</math> is a rational function of the form:

	<math>R(x) = \frac{a_0 + a_1(x - x_0) + a_2 (x - x_0)^2 + \dots + a_m(x - x_0)^m}{1 + b_1(x - x_0) + \dots + b_n (x - x_0)^n}</math>		<math>R(x) = \frac{a_0 + a_1(x - x_0) + a_2 (x - x_0)^2 + \dots + a_m(x - x_0)^m}{1 + b_1(x - x_0) + \dots + b_n (x - x_0)^n}</math>

	where <math>a_i,b_j</math> are all real numbers, and where <math>f^{(j)}(x_0) = R^{(j)}(x_0)</math> for <math>j \in \{ 0, 1,2,\dots,m+n\}</math>.		where <math>a_i,b_j</math> are all real numbers, and where <math>f^{(j)}(x_0) = R^{(j)}(x_0)</math> for <math>j \in \{ 0, 1,2,\dots,m+n\}</math>.

			===About the point 0 and for a given order===

			This definition adapts the previous one for the case <math>x_0 = 0</math>.

			Suppose <math>f</math> is a function and <math>m,n</math> are (possibly equal, possibly distinct) nonnegative integers. Suppose further that <math>f</math> is at least <math>m + n</math> times differentiable at 0.

			The '''Padé approximant''' to <math>f</math> of order <math>[m/n]</math> at 0 is a rational function of the form:

			<math>R(x) = \frac{a_0 + a_1x + a_2x^2 + \dots + a_mx^m}{1 + b_1x + \dots + b_nx^n}</math>

			where <math>a_0,a_1,\dots,a_m,b_1,b_2,\dots,b_n</math> are all real numbers, and where <math>f^{(j)}(0) = R^{(j)}(0)</math> for <math>j \in \{ 0, 1,2,\dots,m+n\}</math>.

Vipul: Created page with "==Definition== ===About a general point and for a given order=== Suppose $f$ is a function, $x_0$ is a point in the domain of $f$ , and
2014-05-02T00:52:43Z

Created page with "==Definition== ===About a general point and for a given order=== Suppose <math>f</math> is a function, <math>x_0</math> is a point in the domain of <math>f</math>, and <math..."

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==Definition==

===About a general point and for a given order===

Suppose <math>f</math> is a function, <math>x_0</math> is a point in the domain of <math>f</math>, and <math>m,n</math> are (possibly equal, possibly distinct) nonnegative integers. The '''Padé approximant''' to <math>f</math> of order <math>[m/n]</math> at <math>x_0</math> is a rational function of the form:

<math>R(x) = \frac{a_0 + a_1(x - x_0) + a_2 (x - x_0)^2 + \dots + a_m(x - x_0)^m}{1 + b_1(x - x_0) + \dots + b_n (x - x_0)^n}</math>

where <math>a_i,b_j</math> are all real numbers, and where <math>f^{(j)}(x_0) = R^{(j)}(x_0)</math> for <math>j \in \{ 0, 1,2,\dots,m+n\}</math>.