Invert an LU Factored Equation

@(@\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}} }@)@ This is cppad-20221105 documentation. Here is a link to its current documentation . 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/utility/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.

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

Input File: include/cppad/utility/lu_invert.hpp