\hypertarget{namespacepolynomial__functions}{}\section{polynomial\+\_\+functions Module Reference} \label{namespacepolynomial__functions}\index{polynomial\+\_\+functions@{polynomial\+\_\+functions}} \subsection{Detailed Description} Polynomial functions. Date of creation\+: 2008.\+06.\+12 L. White. This module contains routines that handle polynomials. \subsection*{Functions/\+Subroutines} \begin{DoxyCompactItemize} \item real function, public \hyperlink{namespacepolynomial__functions_adb2b5d18db527314545e8e21638a2872}{evaluation\+\_\+polynomial} (coeff, ncoef, x) \begin{DoxyCompactList}\small\item\em Pointwise evaluation of a polynomial at x. \end{DoxyCompactList}\item real function, public \hyperlink{namespacepolynomial__functions_a38462b1bc63d3f1f441e4d340c2b4627}{first\+\_\+derivative\+\_\+polynomial} (coeff, ncoef, x) \begin{DoxyCompactList}\small\item\em Calculates the first derivative of a polynomial evaluated at a point x. \end{DoxyCompactList}\item real function, public \hyperlink{namespacepolynomial__functions_ae7112cbad01d6a9477bbaf4a9695ed7a}{integration\+\_\+polynomial} (xi0, xi1, Coeff, npoly) \begin{DoxyCompactList}\small\item\em Exact integration of polynomial of degree npoly. \end{DoxyCompactList}\end{DoxyCompactItemize} \subsection{Function/\+Subroutine Documentation} \mbox{\Hypertarget{namespacepolynomial__functions_adb2b5d18db527314545e8e21638a2872}\label{namespacepolynomial__functions_adb2b5d18db527314545e8e21638a2872}} \index{polynomial\+\_\+functions@{polynomial\+\_\+functions}!evaluation\+\_\+polynomial@{evaluation\+\_\+polynomial}} \index{evaluation\+\_\+polynomial@{evaluation\+\_\+polynomial}!polynomial\+\_\+functions@{polynomial\+\_\+functions}} \subsubsection{\texorpdfstring{evaluation\+\_\+polynomial()}{evaluation\_polynomial()}} {\footnotesize\ttfamily real function, public polynomial\+\_\+functions\+::evaluation\+\_\+polynomial (\begin{DoxyParamCaption}\item[{real, dimension(\+:), intent(in)}]{coeff, }\item[{integer, intent(in)}]{ncoef, }\item[{real, intent(in)}]{x }\end{DoxyParamCaption})} Pointwise evaluation of a polynomial at x. The polynomial is defined by the coefficients contained in the array of the same name, as follows\+: C(1) + C(2)x + C(3)x$^\wedge$2 + C(4)x$^\wedge$3 + ... where C refers to the array \textquotesingle{}coeff\textquotesingle{}. The number of coefficients is given by ncoef and x is the coordinate where the polynomial is to be evaluated. \begin{DoxyParams}[1]{Parameters} \mbox{\tt in} & {\em coeff} & The coefficients of the polynomial\\ \hline \mbox{\tt in} & {\em ncoef} & The number of polynomial coefficients\\ \hline \mbox{\tt in} & {\em x} & The position at which to evaluate the polynomial \\ \hline \end{DoxyParams} Definition at line 20 of file polynomial\+\_\+functions.\+F90. \begin{DoxyCode} 20 \textcolor{keywordtype}{real}, \textcolor{keywordtype}{dimension(:)}, \textcolor{keywordtype}{intent(in)} :: coeff\textcolor{comment}{ !< The coefficients of the polynomial} 21 \textcolor{keywordtype}{integer}, \textcolor{keywordtype}{intent(in)} :: ncoef\textcolor{comment}{ !< The number of polynomial coefficients} 22 \textcolor{keywordtype}{real}, \textcolor{keywordtype}{intent(in)} :: x\textcolor{comment}{ !< The position at which to evaluate the polynomial} 23 \textcolor{comment}{! Local variables} 24 \textcolor{keywordtype}{integer} :: k 25 \textcolor{keywordtype}{real} :: f \textcolor{comment}{! value of polynomial at x} 26 27 f = 0.0 28 \textcolor{keywordflow}{do} k = 1,ncoef 29 f = f + coeff(k) * ( x**(k-1) ) 30 \textcolor{keywordflow}{ enddo} 31 32 evaluation\_polynomial = f 33 \end{DoxyCode} \mbox{\Hypertarget{namespacepolynomial__functions_a38462b1bc63d3f1f441e4d340c2b4627}\label{namespacepolynomial__functions_a38462b1bc63d3f1f441e4d340c2b4627}} \index{polynomial\+\_\+functions@{polynomial\+\_\+functions}!first\+\_\+derivative\+\_\+polynomial@{first\+\_\+derivative\+\_\+polynomial}} \index{first\+\_\+derivative\+\_\+polynomial@{first\+\_\+derivative\+\_\+polynomial}!polynomial\+\_\+functions@{polynomial\+\_\+functions}} \subsubsection{\texorpdfstring{first\+\_\+derivative\+\_\+polynomial()}{first\_derivative\_polynomial()}} {\footnotesize\ttfamily real function, public polynomial\+\_\+functions\+::first\+\_\+derivative\+\_\+polynomial (\begin{DoxyParamCaption}\item[{real, dimension(\+:), intent(in)}]{coeff, }\item[{integer, intent(in)}]{ncoef, }\item[{real, intent(in)}]{x }\end{DoxyParamCaption})} Calculates the first derivative of a polynomial evaluated at a point x. The polynomial is defined by the coefficients contained in the array of the same name, as follows\+: C(1) + C(2)x + C(3)x$^\wedge$2 + C(4)x$^\wedge$3 + ... where C refers to the array \textquotesingle{}coeff\textquotesingle{}. The number of coefficients is given by ncoef and x is the coordinate where the polynomial\textquotesingle{}s derivative is to be evaluated. \begin{DoxyParams}[1]{Parameters} \mbox{\tt in} & {\em coeff} & The coefficients of the polynomial\\ \hline \mbox{\tt in} & {\em ncoef} & The number of polynomial coefficients\\ \hline \mbox{\tt in} & {\em x} & The position at which to evaluate the derivative \\ \hline \end{DoxyParams} Definition at line 44 of file polynomial\+\_\+functions.\+F90. \begin{DoxyCode} 44 \textcolor{keywordtype}{real}, \textcolor{keywordtype}{dimension(:)}, \textcolor{keywordtype}{intent(in)} :: coeff\textcolor{comment}{ !< The coefficients of the polynomial} 45 \textcolor{keywordtype}{integer}, \textcolor{keywordtype}{intent(in)} :: ncoef\textcolor{comment}{ !< The number of polynomial coefficients} 46 \textcolor{keywordtype}{real}, \textcolor{keywordtype}{intent(in)} :: x\textcolor{comment}{ !< The position at which to evaluate the derivative} 47 \textcolor{comment}{! Local variables} 48 \textcolor{keywordtype}{integer} :: k 49 \textcolor{keywordtype}{real} :: f \textcolor{comment}{! value of polynomial at x} 50 51 f = 0.0 52 \textcolor{keywordflow}{do} k = 2,ncoef 53 f = f + \textcolor{keywordtype}{REAL}(k-1)*coeff(k) * ( x**(k-2) ) 54 \textcolor{keywordflow}{ enddo} 55 56 first\_derivative\_polynomial = f 57 \end{DoxyCode} \mbox{\Hypertarget{namespacepolynomial__functions_ae7112cbad01d6a9477bbaf4a9695ed7a}\label{namespacepolynomial__functions_ae7112cbad01d6a9477bbaf4a9695ed7a}} \index{polynomial\+\_\+functions@{polynomial\+\_\+functions}!integration\+\_\+polynomial@{integration\+\_\+polynomial}} \index{integration\+\_\+polynomial@{integration\+\_\+polynomial}!polynomial\+\_\+functions@{polynomial\+\_\+functions}} \subsubsection{\texorpdfstring{integration\+\_\+polynomial()}{integration\_polynomial()}} {\footnotesize\ttfamily real function, public polynomial\+\_\+functions\+::integration\+\_\+polynomial (\begin{DoxyParamCaption}\item[{real, intent(in)}]{xi0, }\item[{real, intent(in)}]{xi1, }\item[{real, dimension(\+:), intent(in)}]{Coeff, }\item[{integer, intent(in)}]{npoly }\end{DoxyParamCaption})} Exact integration of polynomial of degree npoly. The array of coefficients (Coeff) must be of size npoly+1. \begin{DoxyParams}[1]{Parameters} \mbox{\tt in} & {\em xi0} & The lower bound of the integral\\ \hline \mbox{\tt in} & {\em xi1} & The lower bound of the integral\\ \hline \mbox{\tt in} & {\em coeff} & The coefficients of the polynomial\\ \hline \mbox{\tt in} & {\em npoly} & The degree of the polynomial \\ \hline \end{DoxyParams} Definition at line 64 of file polynomial\+\_\+functions.\+F90. \begin{DoxyCode} 64 \textcolor{keywordtype}{real}, \textcolor{keywordtype}{intent(in)} :: xi0\textcolor{comment}{ !< The lower bound of the integral} 65 \textcolor{keywordtype}{real}, \textcolor{keywordtype}{intent(in)} :: xi1\textcolor{comment}{ !< The lower bound of the integral} 66 \textcolor{keywordtype}{real}, \textcolor{keywordtype}{dimension(:)}, \textcolor{keywordtype}{intent(in)} :: coeff\textcolor{comment}{ !< The coefficients of the polynomial} 67 \textcolor{keywordtype}{integer}, \textcolor{keywordtype}{intent(in)} :: npoly\textcolor{comment}{ !< The degree of the polynomial} 68 \textcolor{comment}{! Local variables} 69 \textcolor{keywordtype}{integer} :: k 70 \textcolor{keywordtype}{real} :: integral 71 72 integral = 0.0 73 74 \textcolor{keywordflow}{do} k = 1,npoly+1 75 integral = integral + coeff(k) * (xi1**k - xi0**k) / \textcolor{keywordtype}{real}(k) 76 \textcolor{keywordflow}{ enddo} 77 \textcolor{comment}{!} 78 \textcolor{comment}{!One non-answer-changing way of unrolling the above is:} 79 \textcolor{comment}{! k=1} 80 \textcolor{comment}{! integral = integral + Coeff(k) * (xi1**k - xi0**k) / real(k)} 81 \textcolor{comment}{! if (npoly>=1) then} 82 \textcolor{comment}{! k=2} 83 \textcolor{comment}{! integral = integral + Coeff(k) * (xi1**k - xi0**k) / real(k)} 84 \textcolor{comment}{! endif} 85 \textcolor{comment}{! if (npoly>=2) then} 86 \textcolor{comment}{! k=3} 87 \textcolor{comment}{! integral = integral + Coeff(k) * (xi1**k - xi0**k) / real(k)} 88 \textcolor{comment}{! endif} 89 \textcolor{comment}{! if (npoly>=3) then} 90 \textcolor{comment}{! k=4} 91 \textcolor{comment}{! integral = integral + Coeff(k) * (xi1**k - xi0**k) / real(k)} 92 \textcolor{comment}{! endif} 93 \textcolor{comment}{! if (npoly>=4) then} 94 \textcolor{comment}{! k=5} 95 \textcolor{comment}{! integral = integral + Coeff(k) * (xi1**k - xi0**k) / real(k)} 96 \textcolor{comment}{! endif} 97 \textcolor{comment}{!} 98 integration\_polynomial = integral 99 \end{DoxyCode}