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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.


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.


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.


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>


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

      {1+1○⍵}⍠'g' ○2x  ⍝ Gauss-Legendre
      {1+1○⍵}⍠'n' ○2x  ⍝ Newton-Cotes
      ○2x    ⍝ Exact answer

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

      ⎕PP←60 ⋄ ⎕FPC←512
      {÷1+⍵*2}⍠'g' N
      {÷1+⍵*2}⍠'n' N
      {÷1+⍵*2}⍠('n' 64) N
      {÷1+⍵*2}⍠('n' 96) N
      ¯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

Default Algorithm

When the Variant operator is applied to the default algorithm, the first number is used to replace the exponent N=8 in the calculation of the outer loop limit: 12×2*N. Increasing this value (as in ⍠N) 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 ⎕FPC. The Comparison Tolerance values are expressed as negative integers to a power of 10 as in 10*-1+⌊⎕FPC×10⍟2. It may be changed for any particular calculation by supplying a second numeric argument to the Variant operator as in ⍠(N1 N2), where the first number is a positive integer used as the loop limit exponent and the second number is the negative integer power of 10, i.e. it replaces -1+⌊⎕FPC×10⍟2 in the above statement.

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


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:

      a←2 ⋄ (-a){⍵*2}a

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.


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

      {1+1○⍵}○2x ⋄ ○2x

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


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):

      ¯∞ nd ∞

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 [-√0.5, √0.5] is then a quarter of a circle, whose length is

      ⎕FPC←512 ⋄ ⎕PP←60
      ○0.5x     ⍝ Exact length = 1r2p1

The arc length of the Unit Parabola from 0 to 1 is

      (1r2×(√5r16)÷1r4)+1r4×⍟(1r2+√5r16)÷1r4       ⍝ Exact length (from Wikipedia, "Parabola#Arc_length")