AngouriMath
IntegrateNumerically(AngouriMath.Entity,AngouriMath.Entity.Variable,AngouriMath.Entity.Number.Complex,AngouriMath.Entity.Number.Complex,System.Int32)
Method (no overloads)
Summary
The integral of a function of one variable along the segment from from to
to , by the five-point Gauss–Legendre rule on each of
panels equal panels, in double precision. See
DefiniteIntegral(AngouriMath.Entity.Variable,PeterO.Numbers.EDecimal,PeterO.Numbers.EDecimal,System.Int32).
DefiniteIntegral(AngouriMath.Entity.Variable,PeterO.Numbers.EDecimal,PeterO.Numbers.EDecimal,System.Int32).
Remarks
It was the mean of panels + 1 equally weighted samples, the endpoints among them: a first-order
rule, whose error onx^2 over [0, 1] halved with each doubling, 1.67e-4 at 1000 steps,
and printed to a hundred digits. And a convergent integral with a singular endpoint had no value:
sin(x)/x and ln(x) over [0, 1] were NaN. Gauss–Legendre never evaluates an
endpoint, and the five-point rule is exact on each panel for a polynomial of degree nine. It is run on
the panels, twice as many and four times as many: where the differences do not shrink, the integral
diverges at a bound and is NaN, and otherwise the rest of their geometric series is added and is the
error, and the value is rounded to its place, fifteen digits at most. A singularity inside the
segment is not seen, and a symmetric divergent one, as in 1/x over [-1, 1], comes to its principal
value.
https://github.com/asc-community/AngouriMath/issues/1675
rule, whose error on
and printed to a hundred digits. And a convergent integral with a singular endpoint had no value:
endpoint, and the five-point rule is exact on each panel for a polynomial of degree nine. It is run on
the panels, twice as many and four times as many: where the differences do not shrink, the integral
diverges at a bound and is NaN, and otherwise the rest of their geometric series is added and is the
error, and the value is rounded to its place, fifteen digits at most. A singularity inside the
segment is not seen, and a symmetric divergent one, as in 1/x over [-1, 1], comes to its principal
value.
https://github.com/asc-community/AngouriMath/issues/1675
Parameter "expr"
Expression to integrate
Parameter "x"
Variable to integrate over
Parameter "from"
The complex lower bound for integrating
Parameter "to"
The complex upper bound for integrating
Parameter "panels"
How many panels the segment is cut into, five evaluations each
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