From 8e1a8268e2453dcf6afa6b53ae9bc2b46dbd6d20 Mon Sep 17 00:00:00 2001
From: Olivier Couet <olivier.couet@cern.ch>
Date: Wed, 22 Jun 2016 13:21:17 +0200
Subject: [PATCH] - Apply patches suggested by Mattias Ellert - Remove trailing
white spaces - Spell check
---
math/mathcore/inc/Math/GaussIntegrator.h | 81 ++++++++++++------------
1 file changed, 39 insertions(+), 42 deletions(-)
diff --git a/math/mathcore/inc/Math/GaussIntegrator.h b/math/mathcore/inc/Math/GaussIntegrator.h
index 5c5189b5f7b..d90b47dbf0d 100644
--- a/math/mathcore/inc/Math/GaussIntegrator.h
+++ b/math/mathcore/inc/Math/GaussIntegrator.h
@@ -48,7 +48,7 @@ public:
/** Destructor */
virtual ~GaussIntegrator();
- /** Default Constructor.
+ /** Default Constructor.
If the tolerance are not given, use default values specified in ROOT::Math::IntegratorOneDimOptions
*/
GaussIntegrator(double absTol = -1, double relTol = -1);
@@ -92,31 +92,28 @@ public:
Method:
For any interval [a,b] we define g8(a,b) and g16(a,b) to be the 8-point
and 16-point Gaussian quadrature approximations to
- Begin_Latex
- I = #int^{b}_{a} f(x)dx
- End_Latex
+ \f[
+ I = \int^{b}_{a} f(x)dx
+ \f]
and define
- Begin_Latex
- r(a,b) = #frac{#||{g_{16}(a,b)-g_{8}(a,b)}}{1+#||{g_{16}(a,b)}}
- End_Latex
+ \f[
+ r(a,b) = \frac{\left|g_{16}(a,b)-g_{8}(a,b)\right|}{1+\left|g_{16}(a,b)\right|}
+ \f]
Then,
- Begin_Latex
- G = #sum_{i=1}^{k}g_{16}(x_{i-1},x_{i})
- End_Latex
- where, starting with x0 = A and finishing with xk = B,
- the subdivision points xi(i=1,2,...) are given by
- Begin_Latex
- x_{i} = x_{i-1} + #lambda(B-x_{i-1})
- End_Latex
- Begin_Latex
- #lambda
- End_Latex
- is equal to the first member of the
- sequence 1,1/2,1/4,... for which r(xi-1, xi) < EPS.
+ \f[
+ G = \sum_{i=1}^{k}g_{16}(x_{i-1},x_{i})
+ \f]
+ where, starting with \f$x_{0} = A\f$ and finishing with \f$x_{k} = B\f$,
+ the subdivision points \f$x_{i}(i=1,2,...)\f$ are given by
+ \f[
+ x_{i} = x_{i-1} + \lambda(B-x_{i-1})
+ \f]
+ \f$\lambda\f$ is equal to the first member of the
+ sequence 1,1/2,1/4,... for which \f$r(x_{i-1}, x_{i}) < EPS\f$.
If, at any stage in the process of subdivision, the ratio
- Begin_Latex
- q = #||{#frac{x_{i}-x_{i-1}}{B-A}}
- End_Latex
+ \f[
+ q = \left|\frac{x_{i}-x_{i-1}}{B-A}\right|
+ \f]
is so small that 1+0.005q is indistinguishable from 1 to
machine accuracy, an error exit occurs with the function value
set equal to zero.
@@ -131,13 +128,13 @@ public:
|I|>1, and a bound on the absolute error in the case |I|<1. More
precisely, if k is the number of sub-intervals contributing to the
approximation (see Method), and if
- Begin_Latex
- I_{abs} = #int^{B}_{A} #||{f(x)}dx
- End_Latex
+ \f[
+ I_{abs} = \int^{B}_{A} \left|f(x)\right|dx
+ \f]
then the relation
- Begin_Latex
- #frac{#||{G-I}}{I_{abs}+k} < EPS
- End_Latex
+ \f[
+ \frac{\left|G-I\right|}{I_{abs}+k} < EPS
+ \f]
will nearly always be true, provided the routine terminates without
printing an error message. For functions f having no singularities in
the closed interval [A,B] the accuracy will usually be much higher than
@@ -156,9 +153,9 @@ public:
/** Returns Integral of function on an infinite interval.
This function computes, to an attempted specified accuracy, the value of the integral:
- Begin_Latex
- I = #int^{#infinity}_{-#infinity} f(x)dx
- End_Latex
+ \f[
+ I = \int^{\infty}_{-\infty} f(x)dx
+ \f]
Usage:
In any arithmetic expression, this function has the approximate value
of the integral I.
@@ -169,9 +166,9 @@ public:
/** Returns Integral of function on an upper semi-infinite interval.
This function computes, to an attempted specified accuracy, the value of the integral:
- Begin_Latex
- I = #int^{#infinity}_{A} f(x)dx
- End_Latex
+ \f[
+ I = \int^{\infty}_{A} f(x)dx
+ \f]
Usage:
In any arithmetic expression, this function has the approximate value
of the integral I.
@@ -183,9 +180,9 @@ public:
/** Returns Integral of function on a lower semi-infinite interval.
This function computes, to an attempted specified accuracy, the value of the integral:
- Begin_Latex
- I = #int^{B}_{#infinity} f(x)dx
- End_Latex
+ \f[
+ I = \int^{B}_{-\infty} f(x)dx
+ \f]
Usage:
In any arithmetic expression, this function has the approximate value
of the integral I.
@@ -216,7 +213,7 @@ public:
private:
/**
- Integration surrugate method. Return integral of passed function in interval [a,b]
+ Integration surrogate method. Return integral of passed function in interval [a,b]
Derived class (like GaussLegendreIntegrator) can re-implement this method to modify to use
an improved algorithm
*/
@@ -228,14 +225,14 @@ protected:
double fEpsRel; // Relative error.
double fEpsAbs; // Absolute error.
bool fUsedOnce; // Bool value to check if the function was at least called once.
- double fLastResult; // Result from the last stimation.
- double fLastError; // Error from the last stimation.
+ double fLastResult; // Result from the last estimation.
+ double fLastError; // Error from the last estimation.
const IGenFunction* fFunction; // Pointer to function used.
};
/**
- Auxillary inner class for mapping infinite and semi-infinite integrals
+ Auxiliary inner class for mapping infinite and semi-infinite integrals
*/
class IntegrandTransform : public IGenFunction {
public:
--
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