Integral - Revision history

WikiSysop: /* Examples */

2021-08-29T20:04:32Z

Examples

← Older revision		Revision as of 16:04, 29 August 2021
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	</pre></apll>		</pre></apll>

	A [https://en.wikipedia.org/wiki/Normal_distribution Normal Distribution] is defined as <apll>nd←{(¯0.5×⍵2)÷√○2<_x/>}</apll>. Because this instance of the algorithm is scaled properly, integrating it over the entire width from <apll>¯∞</apll> to <apll>∞</apll> yields an answer of <apll>1</apll> (the area under the curve):		A [https://en.wikipedia.org/wiki/Normal_distribution Normal Distribution] is defined as <apll>nd←{(¯0.5×⍵2<_x/>)÷√○2<_x/>}</apll>. Because this instance of the algorithm is scaled properly, integrating it over the entire width from <apll>¯∞</apll> to <apll>∞</apll> yields an answer of <apll>1</apll> (the area under the curve):

	<apll><pre>		<apll><pre>

WikiSysop: /* Examples */

2021-08-29T02:24:44Z

Examples

← Older revision		Revision as of 22:24, 28 August 2021
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	a←2 ⋄ (-a){⍵*2}<_sg/>a		a←2 ⋄ (-a){⍵*2}<_sg/>a
	5.33333333333333333333333333333333333334		5.33333333333333333333333333333333333334
			2<_r/>3×a*3
			16<_r/>3 ⍝ ←→ 5.33333333333333333333333333333333333333333333333333333333333
	</pre></apll>		</pre></apll>

WikiSysop: /* Variants */

2021-08-28T17:30:23Z

Variants

← Older revision		Revision as of 13:30, 28 August 2021
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	<tr>		<tr>
	<td valign="top">Tanh-Sinh</td>		<td valign="top">Tanh-Sinh</td>
	<td><apll><_sg/>⍠'t'</apll> and <apll><_sg/></apll>~~<br/>as it is the default algorithm~~</td>		<td valign="top"><apll><_sg/>⍠'t'</apll> and <apll><_sg/></apll></td>
	<td valign="top">David H. Bailey's ARPREC ARPREC C++ code translated to C</td>		<td valign="top">David H. Bailey's ARPREC ARPREC C++ code<br/>translated to C — it is the default algorithm</td>
	</tr>		</tr>

WikiSysop: /* Variants */

2021-08-28T02:31:50Z

Variants

← Older revision		Revision as of 22:31, 27 August 2021
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	1.5120405040791739263291383891879796566219342815871082643439<_mark>7</_mark>		1.5120405040791739263291383891879796566219342815871082643439<_mark>7</_mark>
	¯3○N ⍝ Exact answer		¯3○N ⍝ Exact answer
			1.51204050407917392632913838918797965662193428158710826434397
	</pre></apll>		</pre></apll>

WikiSysop: /* Numerical Differentiation */

2020-04-07T19:59:07Z

Numerical Differentiation

← Older revision		Revision as of 15:59, 7 April 2020
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	{⍵*2}{4○⍺⍺∂⍵}∫1		{⍵*2}{4○⍺⍺∂⍵}∫1
	1.4789428575445974338279060194339144350716974305950082518812<_mark>1</_mark>		1.4789428575445974338279060194339144350716974305950082518812<_mark>1</_mark>
	(1<_r/>2×(√5<_r/>16)÷1<_r/>4)+1<_r/>4×⍟(1<_r/>2+√5<_r/>16)÷1<_r/>4 ⍝ Exact length (from Wikipedia, "Parabola#Arc_length")		(1<_r/>2×(√5<_r/>16)÷1<_r/>4)+1<_r/>4×⍟(1<_r/>2+√5<_r/>16)÷1<_r/>4 ⍝ Exact length (from Wikipedia, "Parabola#Arc_length")
	1.47894285754459743382790601943391443507169743059500825188121		1.47894285754459743382790601943391443507169743059500825188121
	</pre></apll>		</pre></apll>

WikiSysop: /* Numerical Differentiation */

2020-04-07T19:53:49Z

Numerical Differentiation

← Older revision		Revision as of 15:53, 7 April 2020
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	○0.5<_x/> ⍝ Exact length = 1<_r/>2<_pi/>1		○0.5<_x/> ⍝ Exact length = 1<_r/>2<_pi/>1
	1.57079632679489661923132169163975144209858469968755291048747</pre></apll>		1.57079632679489661923132169163975144209858469968755291048747</pre></apll>

			The arc length of the Unit Parabola from <apll>0</apll> to <apll>1</apll> is

			<apll><pre>
			{⍵*2}{4○⍺⍺∂⍵}∫1
			1.4789428575445974338279060194339144350716974305950082518812<_mark>1</_mark>
			(1<_r/>2×(√5<_r/>16)÷1<_r/>4)+1<_r/>4×⍟(1<_r/>2+√5<_r/>16)÷1<_r/>4 ⍝ Exact length (from Wikipedia, "Parabola#Arc_length")
			1.47894285754459743382790601943391443507169743059500825188121
			</pre></apll>

WikiSysop at 20:12, 28 March 2020

2020-03-28T20:12:53Z

← Older revision		Revision as of 16:12, 28 March 2020
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	<apll><pre>		<apll><pre>
	⎕FPC←128		⎕FPC←128
	{1+1○⍵}∫⍠'g' ○2<_x/> ⍝ Gauss-Legendre		{1+1○⍵}<_sg/>⍠'g' ○2<_x/> ⍝ Gauss-Legendre
	6.2831853071795864769252867665590057683<_mark>9</_mark>		6.2831853071795864769252867665590057683<_mark>9</_mark>
	{1+1○⍵}∫⍠'n' ○2<_x/> ⍝ Newton-Cotes		{1+1○⍵}<_sg/>⍠'n' ○2<_x/> ⍝ Newton-Cotes
	6.283185307179586476925286766559005<_mark>7</_mark>1385		6.283185307179586476925286766559005<_mark>7</_mark>1385
	○2<_x/> ⍝ Exact answer		○2<_x/> ⍝ Exact answer
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	N←17<_x/>		N←17<_x/>
	⎕PP←60 ⋄ ⎕FPC←512		⎕PP←60 ⋄ ⎕FPC←512
	{÷1+⍵*2}∫⍠'g' N		{÷1+⍵*2}<_sg/>⍠'g' N
	1.5120405040791739263291383891879796566219342815871082643439<_mark>7</_mark>		1.5120405040791739263291383891879796566219342815871082643439<_mark>7</_mark>
	{÷1+⍵*2}∫⍠'n' N		{÷1+⍵*2}<_sg/>⍠'n' N
	1.512040504079173926329138<_mark>3</_mark>4207540700696171248671126465236073		1.512040504079173926329138<_mark>3</_mark>4207540700696171248671126465236073
	{÷1+⍵*2}∫⍠('n' 64) N		{÷1+⍵*2}<_sg/>⍠('n' 64) N
	1.51204050407917392632913838918797965662193428158<_mark>7</_mark>22152948792		1.51204050407917392632913838918797965662193428158<_mark>7</_mark>22152948792
	{÷1+⍵*2}∫⍠('n' 96) N		{÷1+⍵*2}<_sg/>⍠('n' 96) N
	1.5120405040791739263291383891879796566219342815871082643439<_mark>7</_mark>		1.5120405040791739263291383891879796566219342815871082643439<_mark>7</_mark>
	¯3○N ⍝ Exact answer		¯3○N ⍝ Exact answer
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	</tr>		</tr>
	</table>		</table>
			<br/>

	<p>When the Variant operator is applied to the default algorithm, the first number is used to replace the exponent <apll>N=5</apll> in the loop limit of <apll>12×2*N</apll>.</p>		==Default Algorithm==

			<p>When the Variant operator is applied to the default algorithm, the first number is used to replace the exponent <apll>N=8</apll> in the calculation of the outer loop limit: <apll>12×2N</apll>. Increasing this value (as in <apll><_sg/>⍠N</apll>) provides greater precision at the cost of performance. Additionally, two levels of precision (and via another calculation, Comparison Tolerance) are used in this algorithm, one for certain internal values (e.g., abscissa and weights) and the other for more precise calculations (twice the first precision). By default, the precision for less precise calculations is the current value of <apll>⎕FPC</apll>. The Comparison Tolerance values are expressed as negative integers to a power of <apll>10</apll> as in <apll>10-1+⌊⎕FPC×10⍟2</apll>. It may be changed for any particular calculation by supplying a second numeric argument to the Variant operator as in <apll><_sg/>⍠(N1 N2)</apll>, where the first number is a positive integer used as the loop limit exponent and the second number is the negative integer power of <apll>10</apll>, i.e. it replaces <apll>-1+⌊⎕FPC×10⍟2</apll> in the above statement.</p>

	<p>Not all of these additional algorithm's may be supported in future versions.</p>		<p>Not all of these additional algorithm's may be supported in future versions.</p>
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	<apll><pre>		<apll><pre>
	⎕FPC←128		⎕FPC←128
	a←2 ⋄ (-a){⍵*2}∫a		a←2 ⋄ (-a){⍵*2}<_sg/>a
	5.33333333333333333333333333333333333334		5.33333333333333333333333333333333333334
	</pre></apll>		</pre></apll>
Line 169:		Line 172:

	<apll><pre>		<apll><pre>
	2<_r/>3×(2×a3)-(-a){⍵2}∫a		2<_r/>3×(2×a3)-(-a){⍵2}<_sg/>a
	7.1111111111111111111111111111111111111		7.1111111111111111111111111111111111111
	</pre></apll>		</pre></apll>
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	<apll><pre>		<apll><pre>
	{1○⍵}~~∫○2~~<_x/>		{1○⍵}<_sg/>○2<_x/>
	6.69453502284549420282962009846618284241<_E/>¯42		6.69453502284549420282962009846618284241<_E/>¯42
	</pre></apll>		</pre></apll>

	A [https://en.wikipedia.org/wiki/Normal_distribution Normal Distribution] is defined as <apll>nd←{(¯0.5×⍵2)÷√○2<_x/>}</apll>. Because this instance of the algorithm is scaled properly, integrating it over the entire width from <apll>¯∞</apll> to <apll>∞</apll> yields an answer of <apll>1</apll> (the area under the curve)~~. However, to achieve this result, we need to increase the default loop limit exponent~~:		A [https://en.wikipedia.org/wiki/Normal_distribution Normal Distribution] is defined as <apll>nd←{(¯0.5×⍵2)÷√○2<_x/>}</apll>. Because this instance of the algorithm is scaled properly, integrating it over the entire width from <apll>¯∞</apll> to <apll>∞</apll> yields an answer of <apll>1</apll> (the area under the curve):

	<apll><pre>		<apll><pre>
	¯∞ ~~nd∫ ∞~~		¯∞ nd<_sg/> ∞
	~~1.000000000000002522695227400560226259006~~
	~~¯∞ nd∫⍠10~~ ∞
	1		1
	</pre></apll>		</pre></apll>
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	<apll><pre>		<apll><pre>
	⍪¯1 ¯2 ¯3 ~~nd∫¨~~ 1 2 3		⍪¯1 ¯2 ¯3 nd<_sg/>¨ 1 2 3
	0.682689492137085897170465091264075844958 ⍝ 68%		0.682689492137085897170465091264075844958 ⍝ 68%
	0.954499736103641585599434725666933125059 ⍝ 95%		0.954499736103641585599434725666933125059 ⍝ 95%
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	Note that [[Derivative]], the inverse of this operator, has also been implemented. These two features may be combined in various ways, for example, to calculate [https://en.wikipedia.org/wiki/Arc_length#Finding_arc_lengths_by_integrating Arc Length].		Note that [[Derivative]], the inverse of this operator, has also been implemented. These two features may be combined in various ways, for example, to calculate [https://en.wikipedia.org/wiki/Arc_length#Finding_arc_lengths_by_integrating Arc Length].

	The formula for Arc Length is simply <apll>arclen←{4○⍺⍺∂⍵}</apll> which can be used on any given function. For example, the upper half of a unit circle is represented by the function <apll>{0○⍵}</apll>. The interval ~~between~~ <apll>[-√0.5, √0.5]</apll> is then a quarter of a circle, whose length is		The formula for Arc Length is simply <apll>arclen←{4○⍺⍺∂⍵}</apll> which can be used on any given function. For example, the upper half of a unit circle is represented by the function <apll>{0○⍵}</apll>. The interval <apll>[-√0.5, √0.5]</apll> is then a quarter of a circle, whose length is

	<apll><pre>		<apll><pre>
	⎕FPC←512 ⋄ ⎕PP←60		⎕FPC←512 ⋄ ⎕PP←60
	(-√0.5<_x/>){0○⍵}~~arclen∫√0~~.5<_x/>		(-√0.5<_x/>){0○⍵}arclen<_sg/>√0.5<_x/>
	1.5707963267948966192313216916397514420985846996875529104874<_mark>7</_mark>		1.5707963267948966192313216916397514420985846996875529104874<_mark>7</_mark>
	○0.5<_x/> ⍝ Exact length		○0.5<_x/> ⍝ Exact length = 1<_r/>2<_pi/>1
	1.57079632679489661923132169163975144209858469968755291048747</pre></apll>		1.57079632679489661923132169163975144209858469968755291048747</pre></apll>

WikiSysop at 17:09, 27 March 2020

2020-03-27T17:09:31Z

Show changes

WikiSysop at 03:21, 20 January 2020

2020-01-20T03:21:20Z

← Older revision		Revision as of 23:21, 19 January 2020
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	The symbol chosen for this operator is the Integral Sign (<apl><_sg/></apl>), entered from the keyboard as Alt-’S’ or Alt-Shift-'s' (U+222B), and used in mathematics for Integration.		The symbol chosen for this operator is the Integral Sign (<apl><_sg/></apl>), entered from the keyboard as Alt-’S’ or Alt-Shift-'s' (U+222B), and used in mathematics for Integration.

			==Applications==

			There are many, many applications of Numerical Integration. Here are but a few taken from [https://www.whitman.edu Whitman College]'s Mathematics Department's Calculus Online course:

			<ol>
			<li>[https://www.whitman.edu/mathematics/calculus_online/section09.01.html Area between curves]</li>
			<li>[https://www.whitman.edu/mathematics/calculus_online/section09.02.html Distance, Velocity, Acceleration]</li>
			<li>[https://www.whitman.edu/mathematics/calculus_online/section09.03.html Volume]</li>
			<li>[https://www.whitman.edu/mathematics/calculus_online/section09.04.html Average value of a function]</li>
			<li>[https://www.whitman.edu/mathematics/calculus_online/section09.05.html Work]</li>
			<li>[https://www.whitman.edu/mathematics/calculus_online/section09.06.html Center of Mass]</li>
			<li>[https://www.whitman.edu/mathematics/calculus_online/section09.07.html Kinetic energy; improper integrals]</li>
			<li>[https://www.whitman.edu/mathematics/calculus_online/section09.08.html Probability]</li>
			<li>[https://www.whitman.edu/mathematics/calculus_online/section09.09.html Arc Length]</li>
			<li>[https://www.whitman.edu/mathematics/calculus_online/section09.10.html Surface Area]</li>
			</ol>

			and more taken from the website [https://www.intmath.com Interactive Mathematics]:

			<ol>
			<li>[https://www.intmath.com/applications-integration/1-apps-indefinite-integral.php Applications of the Indefinite Integral]</li>
			<li>[https://www.intmath.com/applications-integration/2-area-under-curve.php Area Under a Curve by Integration]</li>
			<li>[https://www.intmath.com/applications-integration/3-area-between-curves.php Area Between 2 Curves using Integration]</li>
			<li>[https://www.intmath.com/applications-integration/4-volume-solid-revolution.php Volume of Solid of Revolution by Integration]</li>
			<li>[https://www.intmath.com/applications-integration/shell-method-volume-solid-revolution.php Shell Method: Volume of Solid of Revolution]</li>
			<li>[https://www.intmath.com/applications-integration/5-centroid-area.php Centroid of an Area by Integration]</li>
			<li>[https://www.intmath.com/applications-integration/6-moments-inertia.php Moments of Inertia by Integration]</li>
			<li>[https://www.intmath.com/applications-integration/7-work-variable-force.php Work by a Variable Force using Integration]</a></li>
			<li>[https://www.intmath.com/applications-integration/8-electric-charges.php Electric Charges by Integration]</li>
			<li>[https://www.intmath.com/applications-integration/9-average-value-function.php Average Value of a Function by Integration]</li>
			<li>[https://www.intmath.com/applications-integration/10-force-liquid.php Force Due to Liquid Pressure by Integration]</a></li>
			<li>[https://www.intmath.com/applications-integration/11-arc-length-curve.php Arc Length of a Curve using Integration]</a></li>
			<li>[https://www.intmath.com/applications-integration/12-arc-length-curve-parametric-polar.php Arc Length of Curve: Parametric, Polar Coordinates]</a></li>
			</ol>

	==Variants==		==Variants==

WikiSysop: /* Variants */

2020-01-13T06:22:37Z

Variants

← Older revision		Revision as of 02:22, 13 January 2020
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	6.2831853071795864769252867665590057683<_mark>9</_mark>		6.2831853071795864769252867665590057683<_mark>9</_mark>
	{1+1○⍵}∫⍠'n' ○2<_x/> ⍝ Newton-Cotes		{1+1○⍵}∫⍠'n' ○2<_x/> ⍝ Newton-Cotes
	6.~~2831853071795864769252867665~~<_mark>5</_mark>~~899537749~~		6.28318530717958647692528676655900<_mark>5</_mark>80189
	○2<_x/> ⍝ Exact answer		○2<_x/> ⍝ Exact answer
	6.28318530717958647692528676655900576839		6.28318530717958647692528676655900576839
	</pre></apll>		</pre></apll>

	The Order of the Numerical Integration (the number of rectangles used to approximate the result) is, by default, <apll>128</apll>. That number may be changed via the Variant operator as in		The Order of the Numerical Integration (the number of rectangles used to approximate the result) is, by default, <apll>128</apll> for the Gauss-Legendre algorithm, and <apll>64</apll> for Newton-Cotes. That number may be changed via the Variant operator as in

	<apll><pre>		<apll><pre>
	N←17<_x/>		N←17<_x/>
			⎕PP←60 ⋄ ⎕FPC←512
	{÷1+⍵*2}∫⍠'g' N		{÷1+⍵*2}∫⍠'g' N
	1.~~5120405040~~<_mark>7</_mark>~~629183188001289665203330309~~		1.512040504079173926329138389187979656<_mark>6</_mark>3103904910367167450076
	{÷1+⍵*2}∫⍠('g' 60) N		{÷1+⍵*2}∫⍠('g' 150) N
	1.~~5120405040791739~~<_mark>2</_mark>~~727026418786910512212~~		1.512040504079173926329138389187979656621934<_mark>2</_mark>7820023884786854
	{÷1+⍵*2}∫⍠('g' 70) N		{÷1+⍵*2}∫⍠('g' 170) N
	1.~~512040504079173926~~<_mark>3</_mark>~~3179507126314397112~~		1.5120405040791739263291383891879796566219342815871<_mark>0</_mark>682038154
	~~{÷1+⍵*2}∫⍠('g' 80) N~~
	~~1.512040504079173926329<_mark>1</_mark>42042030891168~~
	~~{÷1+⍵*2}∫⍠('g' 90) N~~
	~~1.512040504079173926329138<_mark>3</_mark>9289394666393~~
	¯3○N ⍝ Exact answer		¯3○N ⍝ Exact answer
	1.~~51204050407917392632913838918797965662~~		1.51204050407917392632913838918797965662193428158710826434397
	</pre></apll>		</pre></apll>