DEInt

A struct for performing numerical integration by double-exponential (DE) formula. It is also known as "Tanh-sinh quadrature". In DE formula, the integration (1) is converted to (2). (1) int_{xa}^{xb} f(x) dx (2) int_{ta}^{tb} f(g(t)) g'(t) dt

This struct calculates reusable state in advance. The type of DE formula is automatically decided from the given interval of the integration.

Reference:

struct DEInt (

) {

this(F xa, F xb, Flag!"isExpType" isExpType = No.isExpType, size_t trapN = 0, F ta = F.nan, F tb = F.nan);

void setParams(F xa, F xb, Flag!"isExpType" isExpType = No.isExpType, size_t trapN = 0, F ta = F.nan, F tb = F.nan);

F integrate(scope Fn func);

string type [@property getter];

real xa [@property getter];

real xb [@property getter];

real ta [@property getter];

real tb [@property getter];

size_t trapN [@property getter];

bool isExpType [@property getter];

immutable(F)[] xs [@property getter];

immutable(F)[] weights [@property getter];

}

Constructors

this this(F xa, F xb, Flag!"isExpType" isExpType = No.isExpType, size_t trapN = 0, F ta = F.nan, F tb = F.nan): Initialize an object for computing DE integration.

Members

Functions

integrate F integrate(scope Fn func): Execute integration of func by DE formula.
setParams void setParams(F xa, F xb, Flag!"isExpType" isExpType = No.isExpType, size_t trapN = 0, F ta = F.nan, F tb = F.nan): Initialize an object for computing DE integration.

Properties

isExpType bool isExpType [@property getter]
ta real ta [@property getter]
tb real tb [@property getter]
trapN size_t trapN [@property getter]: Return the set value
type string type [@property getter]: Return type of integration. The First and second charactor of the return value are explain the the starting value or end value of integration is finite value ('F') or infinity 'I'. If the return value has the third character and its value is 'E', the integration is formed as int_{xa}^{xb} f(x) exp(-x) dx
weights immutable(F)[] weights [@property getter]: Return weights of each division points.
xa real xa [@property getter]
xb real xb [@property getter]: Return the set value
xs immutable(F)[] xs [@property getter]: Return division points (computing points) of trapezoidal quadrature for DE formula.

Examples

1 // integration on [0, 1]
2 auto int01 = DEInt!real(0, 1);
3 assert(int01.type == "FF");
4 
5 // int_0^1 x dx = 0.5
6 assert(int01.integrate((real x) => x).approxEqual(0.5));
7 
8 // int_0^1 x^^2 dx = 1/3
9 assert(int01.integrate((real x) => x^^2).approxEqual(1/3.0));
10 
11 
12 // integration on [-inf, inf]
13 auto intII = DEInt!real(-real.infinity, real.infinity);
14 assert(intII.type == "II");
15 
16 // Gaussian integral
17 assert(intII.integrate((real x) => exp(-x^^2)).approxEqual(sqrt(PI)));
18 
19 import std.mathspecial;
20 // integration int_1^inf x * exp(-x) dx = Gamma(2, 1)
21 auto intFI = DEInt!real(1, real.infinity, Yes.isExpType);
22 assert(intFI.type == "FIE");
23 
24 // incomplete gamma function
25 assert(intFI.integrate((real x) => x).approxEqual(gammaIncompleteCompl(2, 1) * gamma(2)));