Re: Algorithmic Differentiation in Octave

[Brad Bell]

On 01/27/2017 01:20 PM, Olaf Till wrote:
... snip ...
> Still can't see it... Say we have matrices A:=[a11, a12; a21, a22] and
> B:=[b11, b12; b21, b22], both A and B somehow depending on a vector of
> a_double. If we overload matrix multiplication (normally done symply
> by typing A*B in Octave), to make it recordable, into scalar
> operations, we compute C:=A*B as c11=a11*b11+a12*b21,
> c12=a11*b12+a12*b22, c21=..., c22=... . We have to return the zero
> order result, so we concatenate c11, ..., c22 into a Matrix C
> e.g. with the Octave command C=[c11, c12; c21, c22], and let the
> overloaded method return C. Since all computed c.. should be of type
> a_vector, the operation should have been recorded by cppad. But if we
> run the record to get the derivatives, in which form will they be
> returned? cppad has only records for the four single values c11, ...,
> c22, it doesn't know that they belong together ... or what?
>
> If you correct the above, please be detailed, so that I can understand
> it although I'm not familiar with cppad.
>
> Olaf

There are two separate steps. One is the recording of function 
evaluation to create an AD function object.
The second is using the function object to evaluate derivatives (of any 
order including order zero which is function values).

The function gets recorded by:
1. first declaring the independent variables, say the vector ax. I use 
ax to denote the fact that its elements have type a_double.
2. Performing Operations to eventually get the vector of dependent 
varialbes ay.
3. Stopping the recording and declaring that the function maps ax -> ay.
For example, suppose that we have overloaded matrix multiplication for 
a_double and we enter

ax = m_cppad.independent(x)
A = [ ax(0), ax(1) ; ax(2), ax(3) ]
B = A * A
ay=m_cppad.vec_a_double(4)
ay(0)=B(1,1)
ay(1)=B(1,2)
ay(2)=B(2,1)
ay(3)=B(2,2)
af = m_cppad.a_fun(ax, ay)

Then the function mapping x to y (which uses matrix multiplication in 
its definition) would have been recorded in the function object af.
Does this help ?