emmy.numerical.quadrature.substitute
U Substitution and Variable Changes
This namespace provides implementations of functions that accept an
integrator and perform a variable change to address some singularity, like
an infinite endpoint, in the definite integral.
The strategies currently implemented were each described by Press, et al. in section 4.4 of 'Numerical Recipes'.
### U Substitution and Variable Changes This namespace provides implementations of functions that accept an `integrator` and perform a variable change to address some singularity, like an infinite endpoint, in the definite integral. The strategies currently implemented were each described by Press, et al. in section 4.4 of ['Numerical Recipes'](http://phys.uri.edu/nigh/NumRec/bookfpdf/f4-4.pdf).
exponential-upperclj/s
(exponential-upper integrate)Implements a change of variables to address an exponentially diverging upper integration endpoint. Use this when the integrand diverges as $\exp{x}$ near the upper endpoint $b$.
Implements a change of variables to address an exponentially diverging upper
integration endpoint. Use this when the integrand diverges as $\exp{x}$ near
the upper endpoint $b$.infinitizeclj/s
(infinitize integrate)Performs a variable substitution targeted at converting a single infinite endpoint of an improper integral evaluation into an (open) endpoint at 0 by applying the following substitution:
$$u(t) = {1 \over t}$$ $$du = {-1 \over t^2}$$
This works when the integrand f falls off at least as fast as $1 \over t^2$
as it approaches the infinite limit.
The returned function requires that a and b have the same sign, ie:
$$ab > 0$$
Transform the bounds with $u(t)$, and cancel the negative sign by changing their order:
$$\int_{a}^{b} f(x) d x=\int_{1 / b}^{1 / a} \frac{1}{t^{2}} f\left(\frac{1}{t}\right) dt$$
References:
- Mathworld, "Improper Integral"
- Press, Numerical Recipes, Section 4.4
Performs a variable substitution targeted at converting a single infinite
endpoint of an improper integral evaluation into an (open) endpoint at 0 by
applying the following substitution:
$$u(t) = {1 \over t}$$ $$du = {-1 \over t^2}$$
This works when the integrand `f` falls off at least as fast as $1 \over t^2$
as it approaches the infinite limit.
The returned function requires that `a` and `b` have the same sign, ie:
$$ab > 0$$
Transform the bounds with $u(t)$, and cancel the negative sign by changing
their order:
$$\int_{a}^{b} f(x) d x=\int_{1 / b}^{1 / a} \frac{1}{t^{2}} f\left(\frac{1}{t}\right) dt$$
References:
- Mathworld, ["Improper Integral"](https://mathworld.wolfram.com/ImproperIntegral.html)
- Press, Numerical Recipes, [Section 4.4](http://phys.uri.edu/nigh/NumRec/bookfpdf/f4-4.pdf)inverse-power-law-lowerclj/s
(inverse-power-law-lower integrate gamma)Implements a change of variables to address a power law singularity at the lower integration endpoint.
An "inverse power law singularity" means that the integrand diverges as
$$(x - a)^{-\gamma}$$
near $x=a$.
References:
- Mathworld, "Improper Integral"
- Press, Numerical Recipes, Section 4.4
- Wikipedia, "Finite-time Singularity"
Implements a change of variables to address a power law singularity at the
lower integration endpoint.
An "inverse power law singularity" means that the integrand diverges as
$$(x - a)^{-\gamma}$$
near $x=a$.
References:
- Mathworld, ["Improper Integral"](https://mathworld.wolfram.com/ImproperIntegral.html)
- Press, Numerical Recipes, [Section 4.4](http://phys.uri.edu/nigh/NumRec/bookfpdf/f4-4.pdf)
- Wikipedia, ["Finite-time Singularity"](https://en.wikipedia.org/wiki/Singularity_(mathematics)#Finite-time_singularity)inverse-power-law-upperclj/s
(inverse-power-law-upper integrate gamma)Implements a change of variables to address a power law singularity at the upper integration endpoint.
An "inverse power law singularity" means that the integrand diverges as
$$(x - a)^{-\gamma}$$
near $x=a$.
References:
- Mathworld, "Improper Integral"
- Press, Numerical Recipes, Section 4.4
- Wikipedia, "Finite-time Singularity"
Implements a change of variables to address a power law singularity at the
upper integration endpoint.
An "inverse power law singularity" means that the integrand diverges as
$$(x - a)^{-\gamma}$$
near $x=a$.
References:
- Mathworld, ["Improper Integral"](https://mathworld.wolfram.com/ImproperIntegral.html)
- Press, Numerical Recipes, [Section 4.4](http://phys.uri.edu/nigh/NumRec/bookfpdf/f4-4.pdf)
- Wikipedia, ["Finite-time Singularity"](https://en.wikipedia.org/wiki/Singularity_(mathematics)#Finite-time_singularity)inverse-sqrt-lowerclj/s
(inverse-sqrt-lower integrate)Implements a change of variables to address an inverse square root singularity at the lower integration endpoint. Use this when the integrand diverges as
$$1 \over {\sqrt{x - a}}$$
near the lower endpoint $a$.
Implements a change of variables to address an inverse square root singularity
at the lower integration endpoint. Use this when the integrand diverges as
$$1 \over {\sqrt{x - a}}$$
near the lower endpoint $a$.inverse-sqrt-upperclj/s
(inverse-sqrt-upper integrate)Implements a change of variables to address an inverse square root singularity at the upper integration endpoint. Use this when the integrand diverges as
$$1 \over {\sqrt{x - b}}$$
near the upper endpoint $b$.
Implements a change of variables to address an inverse square root singularity
at the upper integration endpoint. Use this when the integrand diverges as
$$1 \over {\sqrt{x - b}}$$
near the upper endpoint $b$.cljdoc is a website building & hosting documentation for Clojure/Script libraries
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