Secant method - Revision history

Vipul at 16:47, 26 April 2014

2014-04-26T16:47:25Z

← Older revision		Revision as of 16:47, 26 April 2014
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	==Iterative step==		==Iterative step==

	For <math>n \ge 2</math>, we define <math>x_n</math> as the following affine combination of the previous two guesses <math>x_{n-1}</math> and <math>x_{n-2}</math>		For <math>n \ge 2</math>, we define <math>x_n</math> as the following affine combination of the previous two guesses <math>x_{n-1}</math> and <math>x_{n-2}</math>:

	<math>x_n = \frac{x_{n-2}f(x_{n-1}) - x_{n-1}f(x_{n-2})}{f(x_{n-1}) - f(x_{n-2})}</math>		<math>x_n := \frac{x_{n-2}f(x_{n-1}) - x_{n-1}f(x_{n-2})}{f(x_{n-1}) - f(x_{n-2})}</math>

	We can therefore think of <math>x_n</math> as an ''affine linear combination'' of <math>x_{n-2}</math> and <math>x_{n-1}</math> with the following respective coefficients:		We can therefore think of <math>x_n</math> as an ''affine linear combination'' of <math>x_{n-2}</math> and <math>x_{n-1}</math> with the following respective coefficients:
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	Geometrically, this can be interpreted as follows: we make a line through the points <math>(x_{n-2},f(x_{n-2}))</math> and <math>(x_{n-1},f(x_{n-1}))</math> in the <math>(x,f(x))</math>-plane, and define <math>x_n</math> as the <math>x</math>-coordinate of the intersection of this line with the <math>x</math>-axis.		Geometrically, this can be interpreted as follows: we make a line through the points <math>(x_{n-2},f(x_{n-2}))</math> and <math>(x_{n-1},f(x_{n-1}))</math> in the <math>(x,f(x))</math>-plane, and define <math>x_n</math> as the <math>x</math>-coordinate of the intersection of this line with the <math>x</math>-axis.

			==Convergence rate==

Vipul: /* Iterative step */

2014-04-26T16:29:22Z

Iterative step

← Older revision		Revision as of 16:29, 26 April 2014
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	<math>x_n = \frac{x_{n-2}f(x_{n-1}) - x_{n-1}f(x_{n-2})}{f(x_{n-1}) - f(x_{n-2})}</math>		<math>x_n = \frac{x_{n-2}f(x_{n-1}) - x_{n-1}f(x_{n-2})}{f(x_{n-1}) - f(x_{n-2})}</math>

	We can therefore think of <math>x_n</math> as an ''affine linear combination of <math>x_{n-2}</math> and <math>x_{n-1}</math> with the following respective coefficients:		We can therefore think of <math>x_n</math> as an ''affine linear combination'' of <math>x_{n-2}</math> and <math>x_{n-1}</math> with the following respective coefficients:

	<math>\frac{f(x_{n-1})}{f(x_{n-1}) - f(x_{n-2})}, \frac{-f(x_{n-2})}{f(x_{n-1}) - f(x_{n-2})}</math>		<math>\frac{f(x_{n-1})}{f(x_{n-1}) - f(x_{n-2})}, \frac{-f(x_{n-2})}{f(x_{n-1}) - f(x_{n-2})}</math>

	Geometrically, this can be interpreted as follows: we make a line through the points <math>(x_{n-2},f(x_{n-2}))</math> and <math>(x_{n-1},f(x_{n-1}))</math> in the <math>(x,f(x))</math>-plane, and define <math>x_n</math> as the <math>x</math>-coordinate of the intersection of this line with the <math>x</math>-axis.		Geometrically, this can be interpreted as follows: we make a line through the points <math>(x_{n-2},f(x_{n-2}))</math> and <math>(x_{n-1},f(x_{n-1}))</math> in the <math>(x,f(x))</math>-plane, and define <math>x_n</math> as the <math>x</math>-coordinate of the intersection of this line with the <math>x</math>-axis.

Vipul: /* Iterative process */

2014-04-26T16:28:56Z

Iterative process

← Older revision		Revision as of 16:28, 26 April 2014
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	\|}		\|}

	==~~Iterative process~~==		==Initial condition==

	The secant method requires ''two'' initial guesses for the root, say <math>x_0</math> and <math>x_1</math>. For <math>n \ge 2</math>, we define <math>x_n</math> as the following affine combination of the previous two guesses <math>x_{n-1}</math> and <math>x_{n-2}</math>		The secant method requires ''two'' initial guesses for the root, say <math>x_0</math> and <math>x_1</math>.

			==Iterative step==

			For <math>n \ge 2</math>, we define <math>x_n</math> as the following affine combination of the previous two guesses <math>x_{n-1}</math> and <math>x_{n-2}</math>

	<math>x_n = \frac{x_{n-2}f(x_{n-1}) - x_{n-1}f(x_{n-2})}{f(x_{n-1}) - f(x_{n-2})}</math>		<math>x_n = \frac{x_{n-2}f(x_{n-1}) - x_{n-1}f(x_{n-2})}{f(x_{n-1}) - f(x_{n-2})}</math>

			We can therefore think of <math>x_n</math> as an ''affine linear combination of <math>x_{n-2}</math> and <math>x_{n-1}</math> with the following respective coefficients:

			<math>\frac{f(x_{n-1})}{f(x_{n-1}) - f(x_{n-2})}, \frac{-f(x_{n-2})}{f(x_{n-1}) - f(x_{n-2})}</math>

	Geometrically, this can be interpreted as follows: we make a line through the points <math>(x_{n-2},f(x_{n-2}))</math> and <math>(x_{n-1},f(x_{n-1}))</math> in the <math>(x,f(x))</math>-plane, and define <math>x_n</math> as the <math>x</math>-coordinate of the intersection of this line with the <math>x</math>-axis.		Geometrically, this can be interpreted as follows: we make a line through the points <math>(x_{n-2},f(x_{n-2}))</math> and <math>(x_{n-1},f(x_{n-1}))</math> in the <math>(x,f(x))</math>-plane, and define <math>x_n</math> as the <math>x</math>-coordinate of the intersection of this line with the <math>x</math>-axis.

Vipul at 06:21, 26 April 2014

2014-04-26T06:21:42Z

← Older revision		Revision as of 06:21, 26 April 2014
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	The '''secant method''' is a root-finding algorithm that makes successive point estimates for the value of a root of a [[continuous function]]. In general, the secant method is not guaranteed to converge towards a root, but under some conditions, it does. A slight variant of this method, called the [[false position method]], functions very similarly to the [[bisection method]].		The '''secant method''' is a root-finding algorithm that makes successive point estimates for the value of a root of a [[continuous function]]. In general, the secant method is not guaranteed to converge towards a root, but under some conditions, it does. A slight variant of this method, called the [[false position method]], functions very similarly to the [[bisection method]].

			==Summary==

			{\| class="sortable" border="1"
			! Item !! Value
			\|-
			\| Initial condition \|\| we are given a function, typically a [[continuous function]], and two initial guesses <math>x_0</math> and <math>x_1</math> for roots of the function.
			\|-
			\| Iterative step \|\| At stage <math>n</math> (<math>n \ge 2</math>) we compute <math>x_n</math> in terms of <math>x_{n-1}, x_{n-2}, f(x_{n-1}), f(x_{n-2})</math>.
			\|-
			\| Convergence rate \|\| The [[order of convergence]] is the [[golden ratio]]
			\|-
			\| Computational tools needed \|\| Function evaluation at particular points, multiplication, subtraction, and division
			\|-
			\| Termination \|\| We may terminate based on known bounds on the size of the derivative and the function value coming in sufficient proximity to the actual value.
			\|}

	==Iterative process==		==Iterative process==

Vipul at 01:52, 24 April 2014

2014-04-24T01:52:16Z

← Older revision		Revision as of 01:52, 24 April 2014
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	<math>x_n = \frac{x_{n-2}f(x_{n-1}) - x_{n-1}f(x_{n-2})}{f(x_{n-1}) - f(x_{n-2})}</math>		<math>x_n = \frac{x_{n-2}f(x_{n-1}) - x_{n-1}f(x_{n-2})}{f(x_{n-1}) - f(x_{n-2})}</math>

	Geometrically, this can be interpreted as follows: we make a line through the points <math>(x_{n-2},f(x_{n-2})</math> and <math>(x_{n-1},f(x_{n-1})</math> in the <math>(x,f(x))</math>-plane, and define <math>x_n</math> as the <math>x</math>-coordinate of the intersection of this line with the <math>x</math>-axis.		Geometrically, this can be interpreted as follows: we make a line through the points <math>(x_{n-2},f(x_{n-2}))</math> and <math>(x_{n-1},f(x_{n-1}))</math> in the <math>(x,f(x))</math>-plane, and define <math>x_n</math> as the <math>x</math>-coordinate of the intersection of this line with the <math>x</math>-axis.

Vipul: /* =Iterative process */

2014-04-24T01:45:19Z

=Iterative process

← Older revision		Revision as of 01:45, 24 April 2014
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	The '''secant method''' is a root-finding algorithm that makes successive point estimates for the value of a root of a [[continuous function]]. In general, the secant method is not guaranteed to converge towards a root, but under some conditions, it does. A slight variant of this method, called the [[false position method]], functions very similarly to the [[bisection method]].		The '''secant method''' is a root-finding algorithm that makes successive point estimates for the value of a root of a [[continuous function]]. In general, the secant method is not guaranteed to converge towards a root, but under some conditions, it does. A slight variant of this method, called the [[false position method]], functions very similarly to the [[bisection method]].

	==Iterative process=		==Iterative process==

	The secant method requires ''two'' initial guesses for the root, say <math>x_0</math> and <math>x_1</math>. For <math>n \ge 2</math>, we define <math>x_n</math> as the following affine combination of the previous two guesses <math>x_{n-1}</math> and <math>x_{n-2}</math>		The secant method requires ''two'' initial guesses for the root, say <math>x_0</math> and <math>x_1</math>. For <math>n \ge 2</math>, we define <math>x_n</math> as the following affine combination of the previous two guesses <math>x_{n-1}</math> and <math>x_{n-2}</math>

Vipul at 01:45, 24 April 2014

2014-04-24T01:45:06Z

← Older revision		Revision as of 01:45, 24 April 2014
Line 10:		Line 10:
	<math>x_n = \frac{x_{n-2}f(x_{n-1}) - x_{n-1}f(x_{n-2})}{f(x_{n-1}) - f(x_{n-2})}</math>		<math>x_n = \frac{x_{n-2}f(x_{n-1}) - x_{n-1}f(x_{n-2})}{f(x_{n-1}) - f(x_{n-2})}</math>

	Geometrically, this can be interpreted as follows: we make a line through the points <math>(x_{n-2},f(x_[n-2})</math> and <math>(x_{n-1},f(x_{n-1})</math> in the <math>(x,f(x))</math>-plane, and define <math>x_n</math> as the <math>x</math>-coordinate of the intersection of this line with the <math>x</math>-axis.		Geometrically, this can be interpreted as follows: we make a line through the points <math>(x_{n-2},f(x_{n-2})</math> and <math>(x_{n-1},f(x_{n-1})</math> in the <math>(x,f(x))</math>-plane, and define <math>x_n</math> as the <math>x</math>-coordinate of the intersection of this line with the <math>x</math>-axis.

Vipul at 01:44, 24 April 2014

2014-04-24T01:44:18Z

← Older revision		Revision as of 01:44, 24 April 2014
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	==Definition==		==Definition==

	The '''secant method''' is a root-finding algorithm.		The '''secant method''' is a root-finding algorithm that makes successive point estimates for the value of a root of a [[continuous function]]. In general, the secant method is not guaranteed to converge towards a root, but under some conditions, it does. A slight variant of this method, called the [[false position method]], functions very similarly to the [[bisection method]].

			==Iterative process=

			The secant method requires ''two'' initial guesses for the root, say <math>x_0</math> and <math>x_1</math>. For <math>n \ge 2</math>, we define <math>x_n</math> as the following affine combination of the previous two guesses <math>x_{n-1}</math> and <math>x_{n-2}</math>

			<math>x_n = \frac{x_{n-2}f(x_{n-1}) - x_{n-1}f(x_{n-2})}{f(x_{n-1}) - f(x_{n-2})}</math>

			Geometrically, this can be interpreted as follows: we make a line through the points <math>(x_{n-2},f(x_[n-2})</math> and <math>(x_{n-1},f(x_{n-1})</math> in the <math>(x,f(x))</math>-plane, and define <math>x_n</math> as the <math>x</math>-coordinate of the intersection of this line with the <math>x</math>-axis.

Vipul: Created page with "{{root-finding algorithm}} ==Definition== The '''secant method''' is a root-finding algorithm."

2014-04-24T01:21:55Z

Created page with "{{root-finding algorithm}} ==Definition== The '''secant method''' is a root-finding algorithm."

New page

{{root-finding algorithm}}
==Definition==

The '''secant method''' is a root-finding algorithm.