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Up-> CppAD AD ADValued atomic atomic_four atomic_four_example atomic_four_mat_mul

Headings-> Syntax Purpose n_left n_middle n_right call_id x y Theory ---..Forward ---..Matrix Argument Scalar Valued Function ---..Reverse Contents

@(@\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 . Atomic Matrix Multiply Class: Example Implementation
Syntax
atomic_mat_mul mat_mul(name)
call_id = mat_mul.set(n_left, n_middle, n_right)
mat_mul.get(call_id, n_left, n_middle, n_right)
mat_mul(call_id, x, y)

Purpose
Construct an atomic operation that computes the matrix product C = A * B .

n_left
This is the row dimension of the matrices A and C . This is an argument (return value) for the set (get) routine.

n_middle
This is the column dimension of the matrix A and row dimension of the matrix B This is an argument (return value) for the set (get) routine.

n_right
This is the column dimension of the matrices B and C . This is an argument (return value) for the set (get) routine.

call_id
This is a return value (argument) for the set (get) routine.

x
We use x to denote the argument to the atomic function. The size of this vector must be
    n = n_left * n_middle + n_middle * n_right
The matrix A is stored in row major order at the beginning of x ; i.e. its (i,k) element is
    A(i,k) = x[ i * n_middle + k]
The matrix B is stored in row major order at the end of x ; i.e. its (k,j) element is
    B(k,j) = x[ n_left * n_middle + k * n_right + j ]

y
We use y to denote the result of the atomic function. The size of this vector must be m = n_middle * n_right . The matrix C is stored in row major order in y ; i.e. its (i,k) element is
    C(i,j) = y[ i * n_right + j]

Theory

Forward
For @(@ k = 0 , \ldots @)@, the k-th order Taylor coefficient @(@ C^{(k)} @)@ is given by @[@ C^{(k)} = \sum_{\ell = 0}^{k} A^{(\ell)} B^{(k-\ell)} @]@

Matrix Argument Scalar Valued Function
Suppose @(@ \bar{F} @)@ is the derivative of the scalar value function @(@ s(F) @)@ with respect to the matrix @(@ F @)@; i.e., @[@ \bar{F}_{i,j} = \frac{ \partial s } { \partial F_{i,j} } @]@ Also suppose that @(@ t @)@ is a scalar valued argument and @[@ F(t) = D(t) E(t) @]@ It follows that @[@ F'(t) = D'(t) E(t) + D(t) E'(t) @]@ @[@ (s \circ F)'(t) = \R{tr} [ \bar{F}^\R{T} F'(t) ] @]@ @[@ = \R{tr} [ \bar{F}^\R{T} D'(t) E(t) ] + \R{tr} [ \bar{F}^\R{T} D(t) E'(t) ] @]@ @[@ = \R{tr} [ E(t) \bar{F}^\R{T} D'(t) ] + \R{tr} [ \bar{F}^\R{T} D(t) E'(t) ] @]@ Letting @(@ E(t) = 0 @)@ and @(@ D(t) = \Delta^{i,j} (t) @)@ (where @(@ \Delta^{i,j} (t) @)@ is the matrix that is zero, except for @(@ i = j @)@ where it is @(@ t @)@) we have @[@ \bar{D}_{i,j} = \frac{ \partial s } { \partial D_{i,j} } = (s \circ F)'(t) = \R{tr} [ E(t) \bar{F}^\R{T} \Delta^{i,j}(1) ] @]@ @[@ \bar{D}_{i,j} = \sum_k D_{j,k} \bar{F}^\R{T}_{k,i} = \sum_k \bar{F}_{i,k} E^\R{T}_{k,j} @]@ @[@ \bar{D} = \bar{F} E^\R{T} @]@ Letting @(@ D(t) = 0 @)@ and @(@ E(t) = \Delta^{i,j} (t) @)@ we have @[@ \bar{E}_{i,j} = \frac{ \partial s } { \partial E_{i,j} } = (s \circ F)'(t) = \R{tr} [ \bar{F}^\R{T} D(t) \Delta^{i,j} ] @]@ @[@ \bar{E}_{i,j} = \sum_k \bar{F}^\R{T}_{j,k} C_{k,i} = \sum_k D^\R{T}_{i,k} \bar{F}_{k,j} @]@ @[@ \bar{E} = D^\R{T} \bar{F} @]@

Reverse
Reverse mode eliminates @(@ C^{(k)} @)@ as follows: for @(@ \ell = 0, \ldots , k @)@, @[@ \bar{A}^{(\ell)} = \bar{A}^{(\ell)} + \bar{C}^{(k)} [ B^{(k-\ell)} ] ^\R{T} @]@ @[@ \bar{B}^{(k-\ell)} = \bar{B}^{(k-\ell)} + [ A^{(\ell)} ]^\R{T} \bar{C}^{(k)} @]@

Contents

atomic_four_mat_mul_implement	Implementing Atomic Matrix Multiply
atomic_four_mat_mul_forward.cpp	Atomic Matrix Multiply Forward Mode: Example and Test
atomic_four_mat_mul_reverse.cpp	Atomic Matrix Multiply Reverse Mode: Example and Test
atomic_four_mat_mul_sparsity.cpp	Atomic Matrix Multiply Sparsity Patterns: Example and Test
atomic_four_mat_mul_rev_depend.cpp	Atomic Matrix Multiply Reverse Dependency: Example and Test
atomic_four_mat_mul_identical_zero.cpp	Atomic Matrix Multiply Identical Zero: Example and Test

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Input File: include/cppad/example/atomic_four/mat_mul/mat_mul.omh