Difference between revisions of "Integral"

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==Introduction==
 
==Introduction==
  
TBD
+
According to [https://en.wikipedia.org/wiki/Numerical_integration Wikipedia], "In analysis, '''numerical integration''' comprises a broad family of algorithms for calculating the numerical value of a definite integral".
<br />
 
<br />
 
<br />
 
 
 
  
 +
This implementation uses several different algorithms to achieve this as well as extends these algorithms to Infinite and Left and Right semi-infinite intervals.
  
 
==Notation==
 
==Notation==
Line 73: Line 70:
 
==Variants==
 
==Variants==
  
There are several different algorithms that may be used for Numerical Integration, two of which are [https://en.wikipedia.org/wiki/Gaussian_quadrature Gauss-Legendre] and [https://en.wikipedia.org/wiki/Newton%E2%80%93Cotes_formulas Newton-Cotes]. The default algorithm is Gauss-Legendre, however, the faster but less accurate Newton-Cotes algorithm may be selected via the Variant operator as in
+
There are several different algorithms that may be used for Numerical Integration; the main (and default) one is based upon the Tanh-Sinh Quadrature code in David H. Bailey's [https://www.davidhbailey.com/dhbsoftware/arprec-2.2.18.tar.gz ARPREC] package.  Other algorithms include [https://en.wikipedia.org/wiki/Gaussian_quadrature Gauss-Legendre] and [https://en.wikipedia.org/wiki/Newton%E2%80%93Cotes_formulas Newton-Cotes] taken from Laurent Fousse's CRQ (Correctly Rounded Quadrature) [https://github.com/lfousse/crq Numerical Integration] code written in MPFR as described in [https://tel.archives-ouvertes.fr/tel-00477243/document this paper].  These algorithm may be selected via the Variant operator as in
  
 
<apll><pre>
 
<apll><pre>
Line 80: Line 77:
 
6.2831853071795864769252867665590057683<_mark>9</_mark>  
 
6.2831853071795864769252867665590057683<_mark>9</_mark>  
 
       {1+1○⍵}∫⍠'n' ○2<_x/>  ⍝ Newton-Cotes
 
       {1+1○⍵}∫⍠'n' ○2<_x/>  ⍝ Newton-Cotes
6.28318530717958647692528676655900<_mark>5</_mark>80189
+
6.283185307179586476925286766559005<_mark>7</_mark>1385
 
       ○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> for the Gauss-Legendre algorithm, and <apll>64</apll> for Newton-Cotes.  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>32</apll> for Newton-Cotes.  That number may be changed via the Variant operator as in
  
 
<apll><pre>
 
<apll><pre>
Line 91: Line 88:
 
       ⎕PP←60 ⋄ ⎕FPC←512
 
       ⎕PP←60 ⋄ ⎕FPC←512
 
       {÷1+⍵*2}∫⍠'g' N
 
       {÷1+⍵*2}∫⍠'g' N
1.512040504079173926329138389187979656<_mark>6</_mark>3103904910367167450076
+
1.5120405040791739263291383891879796566219342815871082643439<_mark>7</_mark>
       {÷1+⍵*2}∫⍠('g' 150) N
+
      {÷1+⍵*2}∫⍠'n' N
1.512040504079173926329138389187979656621934<_mark>2</_mark>7820023884786854
+
1.512040504079173926329138<_mark>3</_mark>4207540700696171248671126465236073
       {÷1+⍵*2}∫⍠('g' 170) N
+
       {÷1+⍵*2}∫⍠('n' 64) N
1.5120405040791739263291383891879796566219342815871<_mark>0</_mark>682038154
+
1.51204050407917392632913838918797965662193428158<_mark>7</_mark>22152948792
 +
       {÷1+⍵*2}∫⍠('n' 96) N
 +
1.5120405040791739263291383891879796566219342815871082643439<_mark>7</_mark>
 
       ¯3○N    ⍝ Exact answer
 
       ¯3○N    ⍝ Exact answer
1.51204050407917392632913838918797965662193428158710826434397
 
 
</pre></apll>
 
</pre></apll>
 +
 +
<p>There are additional choices for algorithms such as</p>
 +
 +
<table border="1" cellpadding="5" cellspacing="0" rules="none" summary="">
 +
<tr>
 +
  <th align="left">Algorithm</th>
 +
  <th align="left">Syntax</th>
 +
  <th align="left">Source</th>
 +
</tr>
 +
 +
<tr>
 +
  <td>Gauss-Legendre</td>
 +
  <td><apll><_sg/>⍠'G'</apll></td>
 +
  <td>David H. Bailey's ARPREC ARPREC C++ code</td>
 +
</tr>
 +
 +
<tr>
 +
  <td>Error Function</td>
 +
  <td><apll><_sg/>⍠'E'</apll></td>
 +
  <td>David H. Bailey's ARPREC ARPREC C++ code</td>
 +
</tr>
 +
 +
<tr>
 +
  <td>Tanh-Sinh</td>
 +
  <td><apll><_sg/>⍠'T'</apll></td>
 +
  <td>David H. Bailey's ARPREC ARPREC C++ code</td>
 +
</tr>
 +
 +
<tr>
 +
  <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">David H. Bailey's ARPREC ARPREC C++ code translated to C</td>
 +
</tr>
 +
 +
<tr>
 +
  <td>Gauss-Legendre</td>
 +
  <td><apll><_sg/>⍠'g'</apll></td>
 +
  <td>Laurent Fousse's CRQ code</td>
 +
</tr>
 +
 +
<tr>
 +
  <td>Newton-Cotes</td>
 +
  <td><apll><_sg/>⍠'n'</apll></td>
 +
  <td>Laurent Fousse's CRQ code</td>
 +
</tr>
 +
 +
<tr>
 +
  <td>Tanh-Sinh</td>
 +
  <td><apll><_sg/>⍠'d'</apll></td>
 +
  <td>Graeme Dennes Excel Spreadsheet code</td>
 +
</tr>
 +
</table>
 +
 +
<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>
 +
 +
<p>Not all of these additional algorithm's may be supported in future versions.</p>
  
 
==Examples==
 
==Examples==
  
For example,
+
The formula for the [https://en.wikipedia.org/wiki/Parabola#Similarity_to_the_unit_parabola Unit Parabola] is <apll>{⍵*2}</apll> whose Integral is <apll>{1<_r/>3×⍵*3}</apll>, and so the Integral of the Unit Parabola from <apll>-a</apll> to <apll>a</apll> is <apll>-/{1<_r/>3×⍵*3}a,-a ←→ 2<_r/>3×a*3</apll>:
  
The Integral of <apll>{⍵*2}</apll> is <apll>{(⍵*3)÷3}</apll>, and so the Integral of that function from <apll>0</apll> to <apll>1</apll> is <apll>÷3</apll>:
+
<apll><pre>
 +
      ⎕FPC←128
 +
      a←2 ⋄ (-a){⍵*2}∫a
 +
5.33333333333333333333333333333333333334
 +
</pre></apll>
 +
 
 +
This may be used to calculate the [https://en.wikipedia.org/wiki/Parabola#Area_enclosed_between_a_parabola_and_a_chord Area Enclosed Between a Parabola and a Chord], the formula for which is two-thirds of the area of the surrounding rectangle of the chord, the line tangent to the Vertex, and the two sides <apll>x=-a</apll> and <apll>x=a</apll>.
 +
 
 +
Because the Unit Parabola curves upward and away from the X-axis, the above Integral returns the area under (i.e. below and outside) the Parabola.  In this case, because we're Integrating from <apll>-a</apll> to <apll>a</apll>, the chord is the line <apll>y={⍵*2} a ←→ y=a*2</apll>.  The top of the rectangle (chord) is of length <apll>a--a ←→ 2×a</apll> and of height <apll>y ←→ a*2</apll>, whose area is <apll>(2×a)×a*2 ←→ 2×a*3</apll>.  Subtracting the result of the Integral from the rectangle's area and multiplying this by two-thirds yields the area enclosed between a chord and a Parabola.
  
 
<apll><pre>
 
<apll><pre>
       ⎕FPC←128
+
       2<_r/>3×(2×a*3)-(-a){⍵*2}∫a
      {⍵*2}∫1
+
7.1111111111111111111111111111111111111
0.333333333333333333333333333333333333334   
 
 
</pre></apll>
 
</pre></apll>
  
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<apll><pre>
 
<apll><pre>
       {1+1○⍵}∫○2<_x/> ⋄ ○2<_x/>
+
       {1+1○⍵}<_sg/>○2<_x/> ⋄ ○2<_x/>
6.28318530717958647692528676655900576839
+
6.283185307179586476925286766559005768<_mark>3</_mark>4
 
6.28318530717958647692528676655900576839
 
6.28318530717958647692528676655900576839
 
</pre></apll>
 
</pre></apll>
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<apll><pre>
 
<apll><pre>
 
       {1○⍵}∫○2<_x/>
 
       {1○⍵}∫○2<_x/>
¯5.12499200004882449667902954237053816394<_E/>¯40
+
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>.  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, this Integration code doesn't handle infinities as yet, so instead we integrate the function over 20 standard deviations on either side with more rectangles used in the approximation to yield a number within rounding error of the correct answer:
+
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:
  
 
<apll><pre>
 
<apll><pre>
       ¯20 nd∫⍠150 20
+
       ¯∞ nd∫ ∞
1.00000000000000000000000000000000000002
+
1.000000000000002522695227400560226259006
 +
      ¯∞ nd∫⍠10 ∞
 +
1
 
</pre></apll>
 
</pre></apll>
  
Line 138: Line 201:
 
<apll><pre>
 
<apll><pre>
 
       ⍪¯1 ¯2 ¯3 nd∫¨ 1 2 3
 
       ⍪¯1 ¯2 ¯3 nd∫¨ 1 2 3
0.682689492137085897170465091264075844955   ⍝ 68%
+
0.682689492137085897170465091264075844958   ⍝ 68%
0.954499736103641585599434725666933125056   ⍝ 95%
+
0.954499736103641585599434725666933125059   ⍝ 95%
 
0.997300203936739810946696370464810045244    ⍝ 99.7%
 
0.997300203936739810946696370464810045244    ⍝ 99.7%
 
</pre></apll>
 
</pre></apll>
Line 147: Line 210:
 
==Numerical Differentiation==
 
==Numerical Differentiation==
  
Note that [[Derivative]], the inverse of this operator, has also been implemented.
+
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].
  
==Acknowledgements==
+
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
  
This feature is entirely based on Laurent Fousse's [https://github.com/lfousse/crq Numerical Integration] code written in MPFR as described in [https://tel.archives-ouvertes.fr/tel-00477243/document this paper].
+
<apll><pre>
 +
      ⎕FPC←512 ⋄ ⎕PP←60
 +
      (-√0.5<_x/>){0○⍵}arclen∫√0.5<_x/>
 +
1.5707963267948966192313216916397514420985846996875529104874<_mark>7</_mark>
 +
      ○0.5<_x/>    ⍝ Exact length
 +
1.57079632679489661923132169163975144209858469968755291048747</pre></apll>

Revision as of 17:09, 27 March 2020

Z←{L} f R returns the definite Integral of the function f between the points L and R.
L is an optional Real numeric singleton which represents the lower bound of the definite Integral. If it is omitted, 0 is used.
R is a Real numeric singleton which represents the upper bound of the definite Integral.
f is an arbitrary monadic function whose argument and result are both Real numeric singletons.

Introduction

According to Wikipedia, "In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral".

This implementation uses several different algorithms to achieve this as well as extends these algorithms to Infinite and Left and Right semi-infinite intervals.

Notation

The symbol chosen for this operator is the Integral Sign (), 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 Whitman College's Mathematics Department's Calculus Online course:

  1. Area between curves
  2. Distance, Velocity, Acceleration
  3. Volume
  4. Average value of a function
  5. Work
  6. Center of Mass
  7. Kinetic energy; improper integrals
  8. Probability
  9. Arc Length
  10. Surface Area

and more taken from the website Interactive Mathematics:

  1. Applications of the Indefinite Integral
  2. Area Under a Curve by Integration
  3. Area Between 2 Curves using Integration
  4. Volume of Solid of Revolution by Integration
  5. Shell Method: Volume of Solid of Revolution
  6. Centroid of an Area by Integration
  7. Moments of Inertia by Integration
  8. Work by a Variable Force using Integration</a>
  9. Electric Charges by Integration
  10. Average Value of a Function by Integration
  11. Force Due to Liquid Pressure by Integration</a>
  12. Arc Length of a Curve using Integration</a>
  13. Arc Length of Curve: Parametric, Polar Coordinates</a>

Variants

There are several different algorithms that may be used for Numerical Integration; the main (and default) one is based upon the Tanh-Sinh Quadrature code in David H. Bailey's ARPREC package. Other algorithms include Gauss-Legendre and Newton-Cotes taken from Laurent Fousse's CRQ (Correctly Rounded Quadrature) Numerical Integration code written in MPFR as described in this paper. These algorithm may be selected via the Variant operator as in

      ⎕FPC←128
      {1+1○⍵}∫⍠'g' ○2x  ⍝ Gauss-Legendre
6.28318530717958647692528676655900576839 
      {1+1○⍵}∫⍠'n' ○2x  ⍝ Newton-Cotes
6.28318530717958647692528676655900571385
      ○2x    ⍝ Exact answer
6.28318530717958647692528676655900576839

The Order of the Numerical Integration (the number of rectangles used to approximate the result) is, by default, 128 for the Gauss-Legendre algorithm, and 32 for Newton-Cotes. That number may be changed via the Variant operator as in

      N←17x
      ⎕PP←60 ⋄ ⎕FPC←512
      {÷1+⍵*2}∫⍠'g' N
1.51204050407917392632913838918797965662193428158710826434397
      {÷1+⍵*2}∫⍠'n' N
1.51204050407917392632913834207540700696171248671126465236073
      {÷1+⍵*2}∫⍠('n' 64) N
1.51204050407917392632913838918797965662193428158722152948792
      {÷1+⍵*2}∫⍠('n' 96) N
1.51204050407917392632913838918797965662193428158710826434397
      ¯3○N    ⍝ Exact answer

There are additional choices for algorithms such as

Algorithm Syntax Source
Gauss-Legendre ⍠'G' David H. Bailey's ARPREC ARPREC C++ code
Error Function ⍠'E' David H. Bailey's ARPREC ARPREC C++ code
Tanh-Sinh ⍠'T' David H. Bailey's ARPREC ARPREC C++ code
Tanh-Sinh ⍠'t' and
as it is the default algorithm
David H. Bailey's ARPREC ARPREC C++ code translated to C
Gauss-Legendre ⍠'g' Laurent Fousse's CRQ code
Newton-Cotes ⍠'n' Laurent Fousse's CRQ code
Tanh-Sinh ⍠'d' Graeme Dennes Excel Spreadsheet code

When the Variant operator is applied to the default algorithm, the first number is used to replace the exponent N=5 in the loop limit of 12×2*N.

Not all of these additional algorithm's may be supported in future versions.

Examples

The formula for the Unit Parabola is {⍵*2} whose Integral is {1r3×⍵*3}, and so the Integral of the Unit Parabola from -a to a is -/{1r3×⍵*3}a,-a ←→ 2r3×a*3:

      ⎕FPC←128
      a←2 ⋄ (-a){⍵*2}∫a
5.33333333333333333333333333333333333334

This may be used to calculate the Area Enclosed Between a Parabola and a Chord, the formula for which is two-thirds of the area of the surrounding rectangle of the chord, the line tangent to the Vertex, and the two sides x=-a and x=a.

Because the Unit Parabola curves upward and away from the X-axis, the above Integral returns the area under (i.e. below and outside) the Parabola. In this case, because we're Integrating from -a to a, the chord is the line y={⍵*2} a ←→ y=a*2. The top of the rectangle (chord) is of length a--a ←→ 2×a and of height y ←→ a*2, whose area is (2×a)×a*2 ←→ 2×a*3. Subtracting the result of the Integral from the rectangle's area and multiplying this by two-thirds yields the area enclosed between a chord and a Parabola.

      2r3×(2×a*3)-(-a){⍵*2}∫a
7.1111111111111111111111111111111111111

The Integral of the 1+Sine function from 0 to ○2 is ○2:

      {1+1○⍵}○2x ⋄ ○2x
6.28318530717958647692528676655900576834
6.28318530717958647692528676655900576839

and the Integral of the Sine function from 0 to ○2 is (essentially) 0

      {1○⍵}∫○2x
6.69453502284549420282962009846618284241E¯42

A Normal Distribution is defined as nd←{(*¯0.5×⍵*2)÷√○2x}. Because this instance of the algorithm is scaled properly, integrating it over the entire width from ¯∞ to yields an answer of 1 (the area under the curve). However, to achieve this result, we need to increase the default loop limit exponent:

      ¯∞ nd∫ ∞
1.000000000000002522695227400560226259006
      ¯∞ nd∫⍠10 ∞
1

Integrating this same function for one, two, and three standard deviations on either side yields the 3-sigma rule of :

      ⍪¯1 ¯2 ¯3 nd∫¨ 1 2 3
0.682689492137085897170465091264075844958    ⍝ 68%
0.954499736103641585599434725666933125059    ⍝ 95%
0.997300203936739810946696370464810045244    ⍝ 99.7%

which describes about how many of the values in a normal distribution lie within one, two, and three standard deviations from the mean.

Numerical Differentiation

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 Arc Length.

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

      ⎕FPC←512 ⋄ ⎕PP←60
      (-√0.5x){0○⍵}arclen∫√0.5x
1.57079632679489661923132169163975144209858469968755291048747
      ○0.5x     ⍝ Exact length
1.57079632679489661923132169163975144209858469968755291048747