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glossary

Headings-> AD Function AD of Base AD Levels Above Base Base Function Base Type Elementary Vector Operation ---..Atomic ---..Sequence ---..Dependent ---..Independent Parameter Sparsity Pattern ---..Vector of Boolean ---..Vector of Sets Tape ---..Active ---..Inactive ---..Independent Variable ---..Dependent Variables Taylor Coefficient Variable

Glossary
AD Function
Given an ADFun object f there is a corresponding AD of Base operation sequence . This operation sequence defines a function $\mathit{F}:{\mathit{B}}^{\mathit{n}}\to {\mathit{B}}^{\mathit{m}}$ where B is the space corresponding to objects of type Base. We refer to $\mathit{F}$ as the AD function corresponding to the operation sequence and to the object f. (See the FunCheck discussion for possible differences between $\mathit{F}(\mathit{x})$ and the algorithm that defined the operation sequence.)

AD of Base
An object is called an AD of Base object its type is either AD<Base> (see default or constructor ) or VecAD<Base>::reference (see VecAD ) for some Base type.

AD Levels Above Base
If Base is a type, the AD levels above Base is the following sequence of types:
     AD<Base> , AD< AD<Base> > , AD< AD< AD<Base> > > , ...

Base Function
A function $\mathit{f}:\mathit{B}\to \mathit{B}$ is referred to as a Base function, if Base is a C++ type that represent elements of the domain and range space of f; i.e. elements of $\mathit{B}$ .

Base Type
If x is an AD<Base> object, Base is referred to as the base type for x.

Elementary Vector
The j-th elementary vector ${\mathit{e}}^{\mathit{j}}\in {\mathit{B}}^{\mathit{m}}$ is defined by $${\mathit{e}}_{\mathit{i}}^{\mathit{j}}=\{\begin{array}{cc}1\hfill & \mathrm{if}\mathit{i}=\mathit{j}\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}$$
Operation

Atomic
An atomic Type operation is an operation that has a Type result and is not made up of other more basic operations.

Sequence
A sequence of atomic Type operations is called a Type operation sequence. A sequence of atomic AD of Base operations is referred to as an AD of Base operation sequence. The abbreviated notation AD operation sequence is often used when it is not necessary to specify the base type.

Dependent
Suppose that x and y are Type objects and the result of
     x < y
has type bool (where Type is not the same as bool). If one executes the following code
     if( x < y )
          y = cos(x);
     else y = sin(x); 
the choice above depends on the value of x and y and the two choices result in a different Type operation sequence. In this case, we say that the Type operation sequence depends on x and y.

Independent
Suppose that i and n are size_t objects, and x[i], y are Type objects, where Type is different from size_t. The Type sequence of operations corresponding to
     y = Type(0);
     for(i = 0; i < n; i++)
          y += x[i];
does not depend on the value of x or y. In this case, we say that the Type operation sequence is independent of y and the elements of x.

Parameter
All Base objects are parameters. An AD<Base> object u is currently a parameter if its value does not depend on the value of an Independent variable vector for an active tape . If u is a parameter, the function Parameter(u) returns true and Variable(u) returns false.

Sparsity Pattern
CppAD describes a sparse matrix as a vector of sets with each vector component corresponding to a row and the elements of the set corresponding to the possibly non-zero columns. A vector of bool can represent a vector of sets using one bit per element. (Some vectors of bool use one byte per element but vectorBool is an example class that uses one bit per element.) The problem is that this representation uses one bit for both the elements that are there and the ones that are not.

A vector of std::set<size_t> does not represent the elements that are not present, but it uses about three size_t values for each element that is present. For example, if size_t uses 32 bits, a vector of std::set<size_t> uses about 100 bits for each element that is present in the vector of sets. Thus, a vector of std::set<size_t> should be more efficient for very sparse matrix representations.

Vector of Boolean
Given a matrix $\mathit{A}\in {\mathit{B}}^{\mathit{m}\times \mathit{n}}$ , a vector of bool $\mathit{B}$ of length $\mathit{m}\times \mathit{n}$ is a sparsity pattern for $\mathit{A}$ if for $\mathit{i}=0,\dots ,\mathit{m}-1$ and $\mathit{j}=0,\dots \mathit{n}-1$ , $${\mathit{A}}_{\mathit{i},\mathit{j}}\ne 0\Rightarrow {\mathit{B}}_{\mathit{i}*\mathit{n}+\mathit{j}}=\mathrm{true}$$ Given two sparsity patterns $\mathit{B}$ and C for a matrix A, we say that B is more efficient than C if B has fewer true elements than C. For example, if $\mathit{A}$ is the identity matrix, $${\mathit{B}}_{\mathit{i}*\mathit{n}+\mathit{j}}=(\mathit{i}=\mathit{j})$$ defines an efficient sparsity pattern for $\mathit{A}$ .

Vector of Sets
Given a matrix $\mathit{A}\in {\mathit{B}}^{\mathit{m}\times \mathit{n}}$ , a vector of sets $\mathit{S}$ of length $\mathit{m}$ is a sparsity pattern for $\mathit{A}$ if for $\mathit{i}=0,\dots ,\mathit{m}-1$ $${\mathit{A}}_{\mathit{i},\mathit{j}}\ne 0\Rightarrow \mathit{j}\in {\mathit{S}}_{\mathit{i}}$$ Given two sparsity patterns $\mathit{S}$ and T for a matrix A, we say that S is more efficient than T if S has fewer elements than T. For example, if $\mathit{A}$ is the identity matrix, $${\mathit{S}}_{\mathit{i}}=\{\mathit{i}\}$$ defines an efficient sparsity pattern for $\mathit{A}$ .

Tape

Active
A new tape is created and becomes active after each call of the form (see Independent )
     Independent(x)
All operations that depend on the elements of x are recorded on this active tape.

Inactive
The operation sequence stored in a tape must be transferred to a function object using the syntax (see ADFun<Base> f(x, y) )
     ADFun<Base> f( x, y)
or using the syntax (see f.Dependent(x, y) )
     f.Dependent( x, y)
After such a transfer, the tape becomes inactive.

Independent Variable
While the tape is active, we refer to the elements of x as the independent variables for the tape. When the tape becomes inactive, the corresponding objects become parameters .

Dependent Variables
While the tape is active, we use the term dependent variables for the tape for any objects whose value depends on the independent variables for the tape. When the tape becomes inactive, the corresponding objects become parameters .

Taylor Coefficient
Suppose $\mathit{X}:\mathit{B}\to {\mathit{B}}^{\mathit{n}}$ is a is $\mathit{p}$ times continuously differentiable function in some neighborhood of zero. For $\mathit{k}=0,\dots ,\mathit{p}$ , we use the column vector ${\mathit{x}}^{(\mathit{k})}\in {\mathit{B}}^{\mathit{n}}$ for the k-th order Taylor coefficient corresponding to $\mathit{X}$ which is defined by $${\mathit{x}}^{(\mathit{k})}=\frac{1}{\mathit{k}!}\frac{{\partial }^{\mathit{k}}}{\partial {\mathit{t}}^{\mathit{k}}}\mathit{X}(0)$$ It follows that $$\mathit{X}(\mathit{t})={\mathit{x}}^{(0)}+{\mathit{x}}^{(1)}\mathit{t}+\cdots +{\mathit{x}}^{(\mathit{p})}{\mathit{t}}^{\mathit{p}}+\mathit{R}(\mathit{t})$$ where the remainder $\mathit{R}(\mathit{t})$ divided by ${\mathit{t}}^{\mathit{p}}$ converges to zero and $\mathit{t}$ goes to zero.

Variable
An AD<Base> object u is a variable if its value depends on an independent variable vector for a currently active tape . If u is a variable, Variable(u) returns true and Parameter(u) returns false. For example, directly after the code sequence
     Independent(x);
     AD<double> u = x[0];
the AD<double> object u is currently a variable. Directly after the code sequence
     Independent(x);
     AD<double> u = x[0];
     u = 5;
u is currently a parameter (not a variable).

Note that we often drop the word currently and just refer to an AD<Base> object as a variable or parameter.

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