Numerical differentiation - Revision history

Vipul: /* Relative precision of the formulas */

2014-05-31T07:07:35Z

Relative precision of the formulas

← Older revision		Revision as of 07:07, 31 May 2014
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	How the trade-off between greater computational cost and greater precision plays out depends on the nature of the function computation cost and how that changes as we make <math>h</math> smaller.		How the trade-off between greater computational cost and greater precision plays out depends on the nature of the function computation cost and how that changes as we make <math>h</math> smaller.

			===In case of uncertainty regarding differentiability or in case of measurement error, we should not use a single difference quotient but should plot multiple points===

			In the case of uncertainty regarding whether the function is indeed differentiable, or in case of concern about measurement error in the value of the derivative, we should consider plotting the value of the function at the point and nearby points and fitting a linear function, or a low-degree Taylor polynomial function, through the points, using polynomial regression if necessary to find the best fit.

			The simplest form of such a check is that the points of the graph corresponding to the domain points <math>x - (h/2), x, x + (h/2)</math> are almost collinear on the graph of <math>f</math>.

	==Application of numerical differentiation to functions of more than one variable==		==Application of numerical differentiation to functions of more than one variable==

Vipul: /* Application of numerical differentiation to functions of more than one variable */

2014-05-31T07:04:11Z

Application of numerical differentiation to functions of more than one variable

← Older revision		Revision as of 07:04, 31 May 2014
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	<math>(x_1,x_2,\dots,x_n), (x_1 + h,x_2,\dots,x_n), (x_1,x_2 + h,\dots,x_n), \dots, (x_1,x_2,\dots,x_n + h)</math>		<math>(x_1,x_2,\dots,x_n), (x_1 + h,x_2,\dots,x_n), (x_1,x_2 + h,\dots,x_n), \dots, (x_1,x_2,\dots,x_n + h)</math>

	Note that it is not necessary to choose the same <math>h</math> in all directions. We could choose different increments <math>h_1,h_2,\dots,h_n</math> in the different directions, and ~~obtain~~:		Note that it is not necessary to choose the same <math>h</math> in all directions. We could choose different increments <math>h_1,h_2,\dots,h_n</math> in the different directions, and use the values at the corresponding points:

	<math>(x_1,x_2,\dots,x_n), (x_1 + h_1,x_2,\dots,x_n), (x_1,x_2 + h_2,\dots,x_n), \dots, (x_1,x_2,\dots,x_n + h_n)</math>		<math>(x_1,x_2,\dots,x_n), (x_1 + h_1,x_2,\dots,x_n), (x_1,x_2 + h_2,\dots,x_n), \dots, (x_1,x_2,\dots,x_n + h_n)</math>

			* Similarly, if we are using backward difference quotients, or a mix of forward and backward difference quotients, we need to compute the function value at a total of <math>n + 1</math> points.
			* If we use central difference quotients, we need a total of <math>2n</math> function value computations to calculate all the partial derivatives.

Vipul: /* Application of numerical differentiation to functions of more than one variable */

2014-05-31T07:02:44Z

Application of numerical differentiation to functions of more than one variable

← Older revision		Revision as of 07:02, 31 May 2014
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	==Application of numerical differentiation to functions of more than one variable==		==Application of numerical differentiation to functions of more than one variable==

	Suppose <math>f</math> is a function of <math>n</math> variables.		Suppose <math>f</math> is a function of <math>n</math> variables. We can use the method of numerical differentiation to calculate the [[partial derivative]]s of <math>f</math>, and hence the [[gradient vector]] of <math>f</math>. The idea is to compute each partial derivative using a difference quotient. Note that:

			* If we use forward difference quotients for all the partial derivatives, we need to evaluate the function at <math>(n + 1)</math> points: the point at which we are making the computation, and <math>n</math> points, each obtained by moving a bit from the starting point in one coordinate direction. Explicitly, if the point at which we are trying to compute the gradient vector is <math>(x_1,x_2,\dots,x_n)</math> and we choose a value of <math>h</math>, we need to calculate the function values at the points:

			<math>(x_1,x_2,\dots,x_n), (x_1 + h,x_2,\dots,x_n), (x_1,x_2 + h,\dots,x_n), \dots, (x_1,x_2,\dots,x_n + h)</math>

			Note that it is not necessary to choose the same <math>h</math> in all directions. We could choose different increments <math>h_1,h_2,\dots,h_n</math> in the different directions, and obtain:

			<math>(x_1,x_2,\dots,x_n), (x_1 + h_1,x_2,\dots,x_n), (x_1,x_2 + h_2,\dots,x_n), \dots, (x_1,x_2,\dots,x_n + h_n)</math>

Vipul at 02:45, 31 May 2014

2014-05-31T02:45:49Z

@@ Line 14: / Line 14: @@
 ==Relative precision of the formulas==
 Suppose that <math>f</math> has a Taylor series around <math>x</math>. In other words, we can expand:
@@ Line 34: / Line 36: @@
 Note that we for the result above to hold, we do not require the function to have a Taylor series; we only need the function to be three or more times continuously differentiable (in fact, a somewhat weaker version holds if the function is only twice continuously differentiable).

Vipul: /* Relative precision of the formulas */

2014-05-31T02:41:05Z

Relative precision of the formulas

← Older revision		Revision as of 02:41, 31 May 2014
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	! Method !! Computed approximate value to <math>f'(x)</math> !! Error term (computed value minus actual value) !! Order of convergence (smallest exponent on <math>h</matH> with nonzero coefficient in Taylor expansion of error) -- higher order is better		! Method !! Computed approximate value to <math>f'(x)</math> !! Error term (computed value minus actual value) !! Order of convergence (smallest exponent on <math>h</matH> with nonzero coefficient in Taylor expansion of error) -- higher order is better
	\|-		\|-
	\| forward [[difference quotient]] <math>\frac{f(x + h) - f(x)}{h}</math> \|\| <math>f'(x) + \frac{h}{2}f''(x) + \frac{h^3}{6}f'''(x) + \dots</math> \|\| <math>\frac{h}{2}f''(x) + \frac{h^2}{6}f'''(x) + \dots</math> \|\| 1		\| forward [[difference quotient]] <math>\frac{f(x + h) - f(x)}{h}</math> \|\| <math>f'(x) + \frac{h}{2}f''(x) + \frac{h^2}{6}f'''(x) + \dots</math> \|\| <math>\frac{h}{2}f''(x) + \frac{h^2}{6}f'''(x) + \dots</math> \|\| 1
	\|-		\|-
	\| backward [[difference quotient]] <math>\frac{f(x) - f(x - h)}{h}</math> \|\| <math>f'(x) - \frac{h}{2}f''(x) + \frac{h^3}{6}f'''(x) + \dots</math> \|\| <math>- \frac{h}{2}f''(x) + \frac{h^2}{6}f'''(x) + \dots</math> \|\| 1		\| backward [[difference quotient]] <math>\frac{f(x) - f(x - h)}{h}</math> \|\| <math>f'(x) - \frac{h}{2}f''(x) + \frac{h^2}{6}f'''(x) + \dots</math> \|\| <math>- \frac{h}{2}f''(x) + \frac{h^2}{6}f'''(x) + \dots</math> \|\| 1
	\|-		\|-
	\| central difference quotient <math>\frac{f(x + (h/2)) - f(x - (h/2))}{h}</math> \|\| <math>f'(x) + \frac{h^2}{12}f'''(x) + \dots</math> \|\| <math>\frac{h^2}{12}f'''(x) + \dots</math> \|\| 2		\| central difference quotient <math>\frac{f(x + (h/2)) - f(x - (h/2))}{h}</math> \|\| <math>f'(x) + \frac{h^2}{12}f'''(x) + \dots</math> \|\| <math>\frac{h^2}{12}f'''(x) + \dots</math> \|\| 2

Vipul: /* Relative precision of the formulas */

2014-05-31T02:40:51Z

Relative precision of the formulas

← Older revision		Revision as of 02:40, 31 May 2014
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	! Method !! Computed approximate value to <math>f'(x)</math> !! Error term (computed value minus actual value) !! Order of convergence (smallest exponent on <math>h</matH> with nonzero coefficient in Taylor expansion of error) -- higher order is better		! Method !! Computed approximate value to <math>f'(x)</math> !! Error term (computed value minus actual value) !! Order of convergence (smallest exponent on <math>h</matH> with nonzero coefficient in Taylor expansion of error) -- higher order is better
	\|-		\|-
	\| forward [[difference quotient]] <math>\frac{f(x + h) - f(x)}{h}</math> \|\| <math>f'(x) + \frac{h}{2}f''(x) + \frac{h^2}{6}f'''(x) + \dots</math> \|\| <math>\frac{h}{2}f''(x) + \frac{h^2}{6}f'''(x) + \dots</math> \|\| 1		\| forward [[difference quotient]] <math>\frac{f(x + h) - f(x)}{h}</math> \|\| <math>f'(x) + \frac{h}{2}f''(x) + \frac{h^3}{6}f'''(x) + \dots</math> \|\| <math>\frac{h}{2}f''(x) + \frac{h^2}{6}f'''(x) + \dots</math> \|\| 1
	\|-		\|-
	\| backward [[difference quotient]] <math>\frac{f(x) - f(x - h)}{h}</math> \|\| <math>f'(x) - \frac{h}{2}f''(x) + \frac{h^2}{6}f'''(x) + \dots</math> \|\| <math>- \frac{h}{2}f''(x) + \frac{h^3}{6}f'''(x) + \dots</math> \|\| 1		\| backward [[difference quotient]] <math>\frac{f(x) - f(x - h)}{h}</math> \|\| <math>f'(x) - \frac{h}{2}f''(x) + \frac{h^3}{6}f'''(x) + \dots</math> \|\| <math>- \frac{h}{2}f''(x) + \frac{h^2}{6}f'''(x) + \dots</math> \|\| 1
	\|-		\|-
	\| central difference quotient <math>\frac{f(x + (h/2)) - f(x - (h/2))}{h}</math> \|\| <math>f'(x) + \frac{h^2}{12}f'''(x) + \dots</math> \|\| <math>\frac{h^2}{12}f'''(x) + \dots</math> \|\| 2		\| central difference quotient <math>\frac{f(x + (h/2)) - f(x - (h/2))}{h}</math> \|\| <math>f'(x) + \frac{h^2}{12}f'''(x) + \dots</math> \|\| <math>\frac{h^2}{12}f'''(x) + \dots</math> \|\| 2

Vipul at 02:40, 31 May 2014

2014-05-31T02:40:10Z

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 | Backward [[difference quotient]] <math>\frac{f(x) - f(x - h)}{h}</math>, comes from the backward difference form of the [[finite difference]] || The left-hand derivative <math>f'_-(x)</math>. If <math>f</math> is differentiable at <math>x</math>, this equals the two-sided derivative <math>f'(x)</math>.
 |-
-| Central difference quotient <math>\frac{f(x + h) - f(x - h)}{2h}</math>, comes from the central difference form of the [[finite difference]] || If <math>f</math> is differentiable at <math>x</math>, this equals the two-sided derivative <math>f'(x)</math>. Otherwise, however, it does not have any direct interpretation as a one-sided derivative of <math>f</math>.
+| Central difference quotient <math>\frac{f(x + (h/2)) - f(x - (h/2))}{h}</math>, comes from the central difference form of the [[finite difference]] || If <math>f</math> is differentiable at <math>x</math>, this equals the two-sided derivative <math>f'(x)</math>. Otherwise, however, it does not have any direct interpretation as a one-sided derivative of <math>f</math>.
 |}

Vipul at 05:42, 9 May 2014

2014-05-09T05:42:25Z

← Older revision		Revision as of 05:42, 9 May 2014
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	! Expression !! Interpretation of limit as <math>h \to 0</math>		! Expression !! Interpretation of limit as <math>h \to 0</math>
	\|-		\|-
	\| Forward [[difference quotient]] <math>\frac{f(x + h) - f(x)}{h}</math> \|\| The right-hand derivative <math>f'_+(x)</math>. If <math>f</math> is differentiable at <math>x</math>, this equals the two-sided derivative <math>f'(x)</math>.		\| Forward [[difference quotient]] <math>\frac{f(x + h) - f(x)}{h}</math>, comes from the forward difference form of the [[finite difference]] \|\| The right-hand derivative <math>f'_+(x)</math>. If <math>f</math> is differentiable at <math>x</math>, this equals the two-sided derivative <math>f'(x)</math>.
	\|-		\|-
	\| Backward [[difference quotient]] <math>\frac{f(x) - f(x - h)}{h}</math> \|\| The left-hand derivative <math>f'_-(x)</math>. If <math>f</math> is differentiable at <math>x</math>, this equals the two-sided derivative <math>f'(x)</math>.		\| Backward [[difference quotient]] <math>\frac{f(x) - f(x - h)}{h}</math>, comes from the backward difference form of the [[finite difference]] \|\| The left-hand derivative <math>f'_-(x)</math>. If <math>f</math> is differentiable at <math>x</math>, this equals the two-sided derivative <math>f'(x)</math>.
	\|-		\|-
	\| Central difference quotient <math>\frac{f(x + h) - f(x - h)}{2h}</math> \|\| If <math>f</math> is differentiable at <math>x</math>, this equals the two-sided derivative <math>f'(x)</math>. Otherwise, however, it does not have any direct interpretation as a one-sided derivative of <math>f</math>.		\| Central difference quotient <math>\frac{f(x + h) - f(x - h)}{2h}</math>, comes from the central difference form of the [[finite difference]] \|\| If <math>f</math> is differentiable at <math>x</math>, this equals the two-sided derivative <math>f'(x)</math>. Otherwise, however, it does not have any direct interpretation as a one-sided derivative of <math>f</math>.
	\|}		\|}

Vipul: Created page with "==Definition== '''Numerical differentiation''' refers to a method for computing the approximate numerical value of the derivative of a function at a point in the domain a..."

2014-05-09T05:14:51Z

Created page with "==Definition== '''Numerical differentiation''' refers to a method for computing the approximate numerical value of the derivative of a function at a point in the domain a..."

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==Definition==

'''Numerical differentiation''' refers to a method for computing the approximate numerical value of the [[derivative]] of a function at a point in the domain as a [[difference quotient]]. Explicitly, the numerical derivative <math>f'(x)</math> of a function <math>f</math> at a point <math>x</math> may be computed using either of these three formulas, for <math>h</math> a sufficiently small positive real number:

{| class="sortable" border="1"
! Expression !! Interpretation of limit as <math>h \to 0</math>
|-
| Forward [[difference quotient]] <math>\frac{f(x + h) - f(x)}{h}</math> || The right-hand derivative <math>f'_+(x)</math>. If <math>f</math> is differentiable at <math>x</math>, this equals the two-sided derivative <math>f'(x)</math>.
|-
| Backward [[difference quotient]] <math>\frac{f(x) - f(x - h)}{h}</math> || The left-hand derivative <math>f'_-(x)</math>. If <math>f</math> is differentiable at <math>x</math>, this equals the two-sided derivative <math>f'(x)</math>.
|-
| Central difference quotient <math>\frac{f(x + h) - f(x - h)}{2h}</math> || If <math>f</math> is differentiable at <math>x</math>, this equals the two-sided derivative <math>f'(x)</math>. Otherwise, however, it does not have any direct interpretation as a one-sided derivative of <math>f</math>.
|}