From f0b420e8cdd3736e64122c64ab6057b19a12bffc Mon Sep 17 00:00:00 2001 From: Elizabeth Hunt Date: Mon, 27 Nov 2023 15:13:34 -0700 Subject: [PATCH] recompile software manual after hw 7 and hw 7 p6 --- doc/software_manual.pdf | Bin 270972 -> 275601 bytes doc/software_manual.tex | 231 +++++++++++++++++++++++++++++----------- homeworks/hw-7.org | 17 ++- homeworks/hw-7.pdf | Bin 0 -> 119259 bytes homeworks/hw-7.tex | 107 +++++++++++++++++++ src/eigen.c | 32 +++--- 6 files changed, 310 insertions(+), 77 deletions(-) create mode 100644 homeworks/hw-7.pdf create mode 100644 homeworks/hw-7.tex diff --git a/doc/software_manual.pdf b/doc/software_manual.pdf index d099736d16f89d4813f1b7ba04d24712d4235d1d..1f539d7a97ac372672807a9ca387ce7197e28c08 100644 GIT binary patch delta 48651 zcmZ6yV~{3I7q#2AZQHhc+O}=mxZAdE+qQe!wrxxs(`TOdi}S^a6OlhEvMQ=F^2f?+ zueIxI1S#BorXtryp`=(UxgRfxqa<8JW&|Gz z1BfaiqF#}fPJ&SgGmRWgI=-`3=7NY>e4|oE4=)j}mR^LA5yJswYbPX*O3Sc_@58dB 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zhwb^q1Dm?5+tNt-)_C~DcXP^Ehe95 zTP@jnGsl= tolerance && (--iter) > 0) { Array_double *eigenvector_2 = m_dot_v(m, eigenvector_1); + Array_double *normalized_eigenvector_2 = + scale_v(eigenvector_2, 1.0 / linf_norm(eigenvector_2)); + free_vector(eigenvector_2); + eigenvector_2 = normalized_eigenvector_2; Array_double *mx = m_dot_v(m, eigenvector_2); double new_lambda = @@ -1156,8 +1160,115 @@ double dominant_eigenvalue(Matrix_double *m, Array_double *v, double tolerance, return lambda; } \end{verbatim} +\subsubsection{\texttt{shift\_inverse\_power\_eigenvalue}} +\label{sec:org7bb6f14} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Name: \texttt{least\_dominant\_eigenvalue} +\item Location: \texttt{src/eigen.c} +\item Input: a pointer to an invertible matrix \texttt{m}, an initial eigenvector guess \texttt{v} (that is non +zero or orthogonal to an eigenvector with the dominant eigenvalue), a \texttt{shift} to act as the +shifted \(\delta\), and \texttt{tolerance} and \texttt{max\_iterations} that act as stop conditions. +\item Output: the eigenvalue closest to \texttt{shift} with the lowest magnitude closest to 0, approximated +with the Inverse Power Iteration Method +\end{itemize} +\begin{verbatim} +double shift_inverse_power_eigenvalue(Matrix_double *m, Array_double *v, + double shift, double tolerance, + size_t max_iterations) { + assert(m->rows == m->cols); + assert(m->rows == v->size); + + Matrix_double *m_c = copy_matrix(m); + for (size_t y = 0; y < m_c->rows; ++y) + m_c->data[y]->data[y] = m_c->data[y]->data[y] - shift; + + double error = tolerance; + size_t iter = max_iterations; + double lambda = shift; + Array_double *eigenvector_1 = copy_vector(v); + + while (error >= tolerance && (--iter) > 0) { + Array_double *eigenvector_2 = solve_matrix_lu_bsubst(m_c, eigenvector_1); + Array_double *normalized_eigenvector_2 = + scale_v(eigenvector_2, 1.0 / linf_norm(eigenvector_2)); + free_vector(eigenvector_2); + + Array_double *mx = m_dot_v(m, normalized_eigenvector_2); + double new_lambda = + v_dot_v(mx, normalized_eigenvector_2) / + v_dot_v(normalized_eigenvector_2, normalized_eigenvector_2); + + error = fabs(new_lambda - lambda); + lambda = new_lambda; + free_vector(eigenvector_1); + eigenvector_1 = normalized_eigenvector_2; + } + + return lambda; +} +\end{verbatim} +\subsubsection{\texttt{least\_dominant\_eigenvalue}} +\label{sec:orgdef7c62} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Name: \texttt{least\_dominant\_eigenvalue} +\item Location: \texttt{src/eigen.c} +\item Input: a pointer to an invertible matrix \texttt{m}, an initial eigenvector guess \texttt{v} (that is non +zero or orthogonal to an eigenvector with the dominant eigenvalue), a \texttt{tolerance} and +\texttt{max\_iterations} that act as stop conditions. +\item Output: the least dominant eigenvalue with the lowest magnitude closest to 0, approximated +with the Inverse Power Iteration Method. +\end{itemize} +\begin{verbatim} +double least_dominant_eigenvalue(Matrix_double *m, Array_double *v, + double tolerance, size_t max_iterations) { + return shift_inverse_power_eigenvalue(m, v, 0.0, tolerance, max_iterations); +} +\end{verbatim} +\subsubsection{\texttt{partition\_find\_eigenvalues}} +\label{sec:orgc68645a} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Name: \texttt{partition\_find\_eigenvalues} +\item Location: \texttt{src/eigen.c} +\item Input: a pointer to an invertible matrix \texttt{m}, a matrix whose rows correspond to initial +eigenvector guesses at each "partition" which is computed from a uniform distribution +between the number of rows this "guess matrix" has and the distance between the least +dominant eigenvalue and the most dominant. Additionally, a \texttt{max\_iterations} and a \texttt{tolerance} +that act as stop conditions. +\item Output: a vector of \texttt{doubles} corresponding to the "nearest" eigenvalue at the midpoint of +each partition, via the given guess of that partition. +\end{itemize} +\begin{verbatim} +Array_double *partition_find_eigenvalues(Matrix_double *m, + Matrix_double *guesses, + double tolerance, + size_t max_iterations) { + assert(guesses->rows >= + 2); // we need at least, the most and least dominant eigenvalues + + double end = dominant_eigenvalue(m, guesses->data[guesses->rows - 1], + tolerance, max_iterations); + double begin = + least_dominant_eigenvalue(m, guesses->data[0], tolerance, max_iterations); + + double delta = (end - begin) / guesses->rows; + Array_double *eigenvalues = InitArrayWithSize(double, guesses->rows, 0.0); + for (size_t i = 0; i < guesses->rows; i++) { + double box_midpoint = ((delta * i) + (delta * (i + 1))) / 2; + + double nearest_eigenvalue = shift_inverse_power_eigenvalue( + m, guesses->data[i], box_midpoint, tolerance, max_iterations); + + eigenvalues->data[i] = nearest_eigenvalue; + } + + return eigenvalues; +} +\end{verbatim} \subsubsection{\texttt{leslie\_matrix}} -\label{sec:org88d4547} +\label{sec:org0637da1} \begin{itemize} \item Author: Elizabeth Hunt \item Name: \texttt{leslie\_matrix} @@ -1165,7 +1276,7 @@ double dominant_eigenvalue(Matrix_double *m, Array_double *v, double tolerance, \item Input: two pointers to \texttt{Array\_double}'s representing the ratio of individuals in an age class \(x\) getting to the next age class \(x+1\) and the number of offspring that individuals in an age class create in age class 0. -\item Output: the leslie matrix generated with the input vectors. +\item Output: the leslie matrix generated from the input vectors. \end{itemize} \begin{verbatim} @@ -1185,12 +1296,12 @@ Matrix_double *leslie_matrix(Array_double *age_class_surivor_ratio, } \end{verbatim} \subsection{Appendix / Miscellaneous} -\label{sec:org925aa32} +\label{sec:orgddf0893} \subsubsection{Data Types} -\label{sec:org37335a1} +\label{sec:org0ec3831} \begin{enumerate} \item \texttt{Line} -\label{sec:orgaf72b30} +\label{sec:org1b45662} \begin{itemize} \item Author: Elizabeth Hunt \item Location: \texttt{inc/types.h} @@ -1203,7 +1314,7 @@ typedef struct Line { } Line; \end{verbatim} \item The \texttt{Array\_} and \texttt{Matrix\_} -\label{sec:org82faf8e} +\label{sec:org1a49c97} \begin{itemize} \item Author: Elizabeth Hunt \item Location: \texttt{inc/types.h} @@ -1234,10 +1345,10 @@ typedef struct { \end{verbatim} \end{enumerate} \subsubsection{Macros} -\label{sec:org1f988ea} +\label{sec:orge905d93} \begin{enumerate} \item \texttt{c\_max} and \texttt{c\_min} -\label{sec:org8b37b18} +\label{sec:org0267cca} \begin{itemize} \item Author: Elizabeth Hunt \item Location: \texttt{inc/macros.h} @@ -1250,7 +1361,7 @@ typedef struct { #define c_min(x, y) (((x) <= (y)) ? (x) : (y)) \end{verbatim} \item \texttt{InitArray} -\label{sec:org04ec2d7} +\label{sec:orgdb147d1} \begin{itemize} \item Author: Elizabeth Hunt \item Location: \texttt{inc/macros.h} @@ -1270,7 +1381,7 @@ typedef struct { }) \end{verbatim} \item \texttt{InitArrayWithSize} -\label{sec:org4aff8f6} +\label{sec:org0b33fa5} \begin{itemize} \item Author: Elizabeth Hunt \item Location: \texttt{inc/macros.h} @@ -1290,7 +1401,7 @@ typedef struct { }) \end{verbatim} \item \texttt{InitMatrixWithSize} -\label{sec:org3457577} +\label{sec:orgcc5a2bb} \begin{itemize} \item Author: Elizabeth Hunt \item Location: \texttt{inc/macros.h} diff --git a/homeworks/hw-7.org b/homeworks/hw-7.org index e18d1ee..2c28af2 100644 --- a/homeworks/hw-7.org +++ b/homeworks/hw-7.org @@ -1,4 +1,4 @@ -#+TITLE: Homework 6 +#+TITLE: Homework 7 #+AUTHOR: Elizabeth Hunt #+LATEX_HEADER: \notindent \notag \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry} #+LATEX: \setlength\parindent{0pt} @@ -58,4 +58,19 @@ See also the entry ~Eigen-Adjacent -> partition_find_eigenvalues~ in the LIZFCM documentation. * Question Six +Consider we have the results of two methods developed in this homework: ~least_dominant_eigenvalue~, and ~dominant_eigenvalue~ +into ~lambda_0~, ~lambda_n~, respectively. Also assume that we have the method implemented as we've introduced, +~shift_inverse_power_eigenvalue~. +Then, we begin at the midpoint of ~lambda_0~ and ~lambda_n~, and compute the +~new_lambda = shift_inverse_power_eigenvalue~ +with a shift at the midpoint, and some given initial guess. + +1. If the result is equal (or within some tolerance) to ~lambda_n~ then the closest eigenvalue to the midpoint + is still the dominant eigenvalue, and thus the next most dominant will be on the left. Set ~lambda_n~ + to the midpoint and reiterate. +2. If the result is greater or equal to ~lambda_0~ we know an eigenvalue of greater or equal magnitude + exists on the right. So, we set ~lambda_0~ to this eigenvalue associated with the midpoint, and + re-iterate. +3. Continue re-iterating until we hit some given maximum number of iterations. 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+\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{capt-of} +\usepackage{hyperref} +\notindent \notag \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry} +\author{Elizabeth Hunt} +\date{\today} +\title{Homework 7} +\hypersetup{ + pdfauthor={Elizabeth Hunt}, + pdftitle={Homework 7}, + pdfkeywords={}, + pdfsubject={}, + pdfcreator={Emacs 29.1 (Org mode 9.7-pre)}, + pdflang={English}} +\begin{document} + +\maketitle +\setlength\parindent{0pt} +\section{Question One} +\label{sec:org8ef0ee6} +See \texttt{UTEST(eigen, dominant\_eigenvalue)} in \texttt{test/eigen.t.c} and the entry +\texttt{Eigen-Adjacent -> dominant\_eigenvalue} in the LIZFCM API documentation. +\section{Question Two} +\label{sec:orgbdba5c1} +See \texttt{UTEST(eigen, leslie\_matrix\_dominant\_eigenvalue)} in \texttt{test/eigen.t.c} +and the entry \texttt{Eigen-Adjacent -> leslie\_matrix} in the LIZFCM API +documentation. +\section{Question Three} +\label{sec:org19b04f4} +See \texttt{UTEST(eigen, least\_dominant\_eigenvalue)} in \texttt{test/eigen.t.c} which +finds the least dominant eigenvalue on the matrix: + +\begin{bmatrix} +2 & 2 & 4 \\ +1 & 4 & 7 \\ +0 & 2 & 6 +\end{bmatrix} + +which has eigenvalues: \(5 + \sqrt{17}, 2, 5 - \sqrt{17}\) and should thus produce \(5 - \sqrt{17}\). + +See also the entry \texttt{Eigen-Adjacent -> least\_dominant\_eigenvalue} in the LIZFCM API +documentation. +\section{Question Four} +\label{sec:orgc58d42d} +See \texttt{UTEST(eigen, shifted\_eigenvalue)} in \texttt{test/eigen.t.c} which +finds the least dominant eigenvalue on the matrix: + +\begin{bmatrix} +2 & 2 & 4 \\ +1 & 4 & 7 \\ +0 & 2 & 6 +\end{bmatrix} + +which has eigenvalues: \(5 + \sqrt{17}, 2, 5 - \sqrt{17}\) and should thus produce \(2.0\). + +With the initial guess: \([0.5, 1.0, 0.75]\). + +See also the entry \texttt{Eigen-Adjacent -> shift\_inverse\_power\_eigenvalue} in the LIZFCM API +documentation. +\section{Question Five} +\label{sec:orga369221} +See \texttt{UTEST(eigen, partition\_find\_eigenvalues)} in \texttt{test/eigen.t.c} which +finds the eigenvalues in a partition of 10 on the matrix: + +\begin{bmatrix} +2 & 2 & 4 \\ +1 & 4 & 7 \\ +0 & 2 & 6 +\end{bmatrix} + +which has eigenvalues: \(5 + \sqrt{17}, 2, 5 - \sqrt{17}\), and should produce all three from +the partitions when given the guesses \([0.5, 1.0, 0.75]\) from the questions above. + +See also the entry \texttt{Eigen-Adjacent -> partition\_find\_eigenvalues} in the LIZFCM API +documentation. +\section{Question Six} +\label{sec:orgadc3078} +Consider we have the results of two methods developed in this homework: \texttt{least\_dominant\_eigenvalue}, and \texttt{dominant\_eigenvalue} +into \texttt{lambda\_0}, \texttt{lambda\_n}, respectively. Also assume that we have the method implemented as we've introduced, +\texttt{shift\_inverse\_power\_eigenvalue}. + +Then, we begin at the midpoint of \texttt{lambda\_0} and \texttt{lambda\_n}, and compute the +\texttt{new\_lambda = shift\_inverse\_power\_eigenvalue} +with a shift at the midpoint, and some given initial guess. + +\begin{enumerate} +\item If the result is equal (or within some tolerance) to \texttt{lambda\_n} then the closest eigenvalue to the midpoint +is still the dominant eigenvalue, and thus the next most dominant will be on the left. Set \texttt{lambda\_n} +to the midpoint and reiterate. +\item If the result is greater or equal to \texttt{lambda\_0} we know an eigenvalue of greater or equal magnitude +exists on the right. So, we set \texttt{lambda\_0} to this eigenvalue associated with the midpoint, and +re-iterate. +\item Continue re-iterating until we hit some given maximum number of iterations. Finally we will return +\texttt{new\_lambda}. +\end{enumerate} +\end{document} diff --git a/src/eigen.c b/src/eigen.c index 92ef88c..49cc0e4 100644 --- a/src/eigen.c +++ b/src/eigen.c @@ -4,19 +4,9 @@ #include #include -Matrix_double *leslie_matrix(Array_double *age_class_surivor_ratio, - Array_double *age_class_offspring) { - assert(age_class_surivor_ratio->size + 1 == age_class_offspring->size); - - Matrix_double *leslie = InitMatrixWithSize(double, age_class_offspring->size, - age_class_offspring->size, 0.0); - - free_vector(leslie->data[0]); - leslie->data[0] = copy_vector(age_class_offspring); - - for (size_t i = 0; i < age_class_surivor_ratio->size; i++) - leslie->data[i + 1]->data[i] = age_class_surivor_ratio->data[i]; - return leslie; +double least_dominant_eigenvalue(Matrix_double *m, Array_double *v, + double tolerance, size_t max_iterations) { + return shift_inverse_power_eigenvalue(m, v, 0.0, tolerance, max_iterations); } double dominant_eigenvalue(Matrix_double *m, Array_double *v, double tolerance, @@ -110,7 +100,17 @@ Array_double *partition_find_eigenvalues(Matrix_double *m, return eigenvalues; } -double least_dominant_eigenvalue(Matrix_double *m, Array_double *v, - double tolerance, size_t max_iterations) { - return shift_inverse_power_eigenvalue(m, v, 0.0, tolerance, max_iterations); +Matrix_double *leslie_matrix(Array_double *age_class_surivor_ratio, + Array_double *age_class_offspring) { + assert(age_class_surivor_ratio->size + 1 == age_class_offspring->size); + + Matrix_double *leslie = InitMatrixWithSize(double, age_class_offspring->size, + age_class_offspring->size, 0.0); + + free_vector(leslie->data[0]); + leslie->data[0] = copy_vector(age_class_offspring); + + for (size_t i = 0; i < age_class_surivor_ratio->size; i++) + leslie->data[i + 1]->data[i] = age_class_surivor_ratio->data[i]; + return leslie; }