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Up-> CppAD utility LuDetAndSolve LuInvert

CppAD-> Install Introduction AD ADFun preprocessor multi_thread utility ipopt_solve Example speed Appendix

utility-> ErrorHandler NearEqual speed_test SpeedTest time_test test_boolofvoid NumericType CheckNumericType SimpleVector CheckSimpleVector nan pow_int Poly LuDetAndSolve RombergOne RombergMul Runge45 Rosen34 OdeErrControl OdeGear OdeGearControl CppAD_vector thread_alloc index_sort to_string set_union sparse_rc sparse_rcv

LuDetAndSolve-> LuSolve LuFactor LuInvert

LuInvert-> lu_invert.cpp lu_invert.hpp

Headings-> Syntax Description Include Matrix Storage ip jp LU ---..L ---..U ---..P ---..A X Example Source

@(@\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }@)@ Invert an LU Factored Equation
Syntax
 include <cppad/utility/lu_invert.hpp>
LuInvert(ip, jp, LU, X)

Description
Solves the matrix equation A * X = B using an LU factorization computed by LuFactor .

Include
The file cppad/lu_invert.hpp is included by cppad/cppad.hpp but it can also be included separately with out the rest of the CppAD routines.

Matrix Storage
All matrices are stored in row major order. To be specific, if @(@ Y @)@ is a vector that contains a @(@ p @)@ by @(@ q @)@ matrix, the size of @(@ Y @)@ must be equal to @(@ p * q @)@ and for @(@ i = 0 , \ldots , p-1 @)@, @(@ j = 0 , \ldots , q-1 @)@, @[@ Y_{i,j} = Y[ i * q + j ] @]@

ip
The argument ip has prototype
     const SizeVector &ip
(see description for SizeVector in LuFactor specifications). The size of ip is referred to as n in the specifications below. The elements of ip determine the order of the rows in the permuted matrix.

jp
The argument jp has prototype
     const SizeVector &jp
(see description for SizeVector in LuFactor specifications). The size of jp must be equal to n . The elements of jp determine the order of the columns in the permuted matrix.

LU
The argument LU has the prototype
     const FloatVector &LU
and the size of LU must equal @(@ n * n @)@ (see description for FloatVector in LuFactor specifications).

L
We define the lower triangular matrix L in terms of LU . The matrix L is zero above the diagonal and the rest of the elements are defined by
     L(i, j) = LU[ ip[i] * n + jp[j] ]
for @(@ i = 0 , \ldots , n-1 @)@ and @(@ j = 0 , \ldots , i @)@.

U
We define the upper triangular matrix U in terms of LU . The matrix U is zero below the diagonal, one on the diagonal, and the rest of the elements are defined by
     U(i, j) = LU[ ip[i] * n + jp[j] ]
for @(@ i = 0 , \ldots , n-2 @)@ and @(@ j = i+1 , \ldots , n-1 @)@.

P
We define the permuted matrix P in terms of the matrix L and the matrix U by P = L * U .

A
The matrix A , which defines the linear equations that we are solving, is given by
     P(i, j) = A[ ip[i] * n + jp[j] ]
(Hence LU contains a permuted factorization of the matrix A .)

X
The argument X has prototype
     FloatVector &X
(see description for FloatVector in LuFactor specifications). The matrix X must have the same number of rows as the matrix A . The input value of X is the matrix B and the output value solves the matrix equation A * X = B .

Example
The file lu_solve.hpp is a good example usage of LuFactor with LuInvert. The file lu_invert.cpp contains an example and test of using LuInvert by itself. It returns true if it succeeds and false otherwise.

Source
The file lu_invert.hpp contains the current source code that implements these specifications.

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Input File: cppad/utility/lu_invert.hpp