10. SYMDIFF

10.1. Overview

SYMDIFF is a tool capable of evaluating derivatives of symbolic expressions. Using a natural syntax, it is possible to manipulate symbolic equations in order to aid derivation of equations for a variety of applications. It has been tailored for use within DEVSIM.

10.2. Syntax

10.2.1. Variables and numbers

Variables and numbers are the basic building blocks for expressions. A variable is defined as any sequence of characters beginning with a letter and followed by letters, integer digits, and the _ character. Note that the letters are case sensitive so that a and {A} are not the same variable. Any other characters are considered to be either mathematical operators or invalid, even if there is no space between the character and the rest of the variable name.

Examples of valid variable names are:

a, dog, var1, var_2

Numbers can be integer or floating point. Scientific notation is accepted as a valid syntax. For example:

1.0, 1.0e-2, 3.4E-4

10.2.2. Basic expressions

Table 10.1 Basic expressions involving unary, binary, and logical operators.

Expression	Description
(exp1)	Parenthesis for changing precedence
+exp1	Unary Plus
-exp1	Unary Minus
!exp1	Logical Not
exp1 ^ exp2	Exponentiation
exp1 * exp2	Multiplication
exp1 / exp2	Division
exp1 + exp2	Addition
exp1 - exp2	Subtraction
exp1 < exp2	Test Less
exp1 <= exp2	Test Less Equal
exp1 > exp2	Test Greater
exp1 >= exp2	Test Greater Equal
exp1 == exp2	Test Equality
exp1 != exp2	Test Inequality
exp1 && exp2	Logical And
exp1 || exp2	Logical Or
variable	Independent Variable
number	Integer or decimal number

In Table 10.1, the basic syntax for the language is presented. An expression may be composed of variables and numbers tied together with mathematical operations. Order of operations is from bottom to top in order of increasing precedence. Operators with the same level of precedence are contained within horizontal lines.

In the expression a + b * c, the multiplication will be performed before the addition. In order to override this precedence, parenthesis may be used. For example, in (a + b) * c, the addition operation is performed before the multiplication.

The logical operators are based on non zero values being true and zero values being false. The test operators are evaluate the numerical values and result in 0 for false and 1 for true.

It is important to note since values are based on double precision arithmetic, testing for equality with values other than 0.0 may yield unexpected results.

10.2.3. Functions

Table 10.2 Predefined Functions

Function	Description
acosh(exp1)	Inverse Hyperbolic Cosine
asinh(exp1)	Inverse Hyperbolic Sine
atanh(exp1)	Inverse Hyperbolic Tangent
cosh(exp1)	Hyperbolic Cosine
sinh(exp1)	Hyperbolic Sine
tanh(exp1)	Hyperbolic Tangent
B(exp1)	Bernoulli Function
dBdx(exp1)	derivative of Bernoulli function
dot2d(exp1x, exp1y, exp2x, exp2y)	exp1x*exp2x+exp1y*exp2y
exp(exp1)	exponent
ifelse(test, exp1, exp2)	if test is true, then evaluate exp1, otherwise exp2
if(test, exp)	if test is true, then evaluate exp, otherwise 0
log(exp1)	natural log
max(exp1, exp2)	maximum of the two arguments
min(exp1, exp2)	minimum of the two arguments
pow(exp1, exp2)	take exp1 to the power of exp2
sgn(exp1)	sign function
step(exp1)	unit step function
kahan3(exp1, exp2, exp3)	Extended precision addition of arguments
kahan4(exp1, exp2, exp3, exp4)	Extended precision addition of arguments
vec_max	maximum of all the values over the entire region or interface
vec_min	minimum of all the values over the entire region or interface
vec_sum	sum of all the values over the entire region or interface

Table 10.3 Error Functions

Function	Description
erfc(exp1)	complementary error function
derfcdx(exp1)	derivative of complementary error function
erfc_inv(exp1)	inverse complementary error function
derfc_invdx(exp1)	derivative of inverse complementary error function
erf(exp1)	error function
derfdx(exp1)	derivative error function
erf_inv(exp1)	inverse error function
derf_invdx(exp1)	derivative of inverse error function

Table 10.4 Fermi Integral Functions

Function	Description
Fermi(exp1)	Fermi Integral
dFermidx(exp1)	derivative of Fermi Integral
InvFermi(exp1)	inverse of the Fermi Integral
dInvFermidx(exp1)	derivative of InvFermi Integral

Table 10.5 Gauss-Fermi Integral Functions

gfi(exp1, exp2)	Gauss-Fermi Integral
dgfidx(exp1, exp2)	Derivative of Gauss-Fermi Integral with respect to first argument
igfi(exp1, exp2)	Inverse Gauss-Fermi Integral
digfidx(exp1, exp2)	Derivative of Inverse Gauss-Fermi Integral with respect to first argument

In Table 10.2 are the built in functions of SYMDIFF. Note that the pow function uses the , operator to separate arguments. In addition an expression like pow(a,b+y) is equivalent to an expression like a^(b+y). Both exp and log are provided since many derivative expressions can be expressed in terms of these two functions. It is possible to nest expressions within functions and vice-versa. Table 10.3 lists the error functions, derivatives, and inverses. Table 10.4 lists the Fermi functions, and are based on the Joyce-Dixon Approximation [4]. The Gauss-Fermi functions are listed in Table 10.5, based on [7].

10.2.4. Commands

Table 10.6 Commands.

Command	Description
diff(obj1, var)	Take derivative of obj1 with respect to variable var
expand(obj)	Expand out all multiplications into a sum of products
help	Print description of commands
scale(obj)	Get constant factor
sign(obj)	Get sign as 1 or -1
simplify(obj)	Simplify as much as possible
subst(obj1,obj2,obj3)	substitute obj3 for obj2 into obj1
unscaledval(obj)	Get value without constant scaling
unsignedval(obj)	Get unsigned value

Commands are shown in Table 10.6. While they appear to have the same form as functions, they are special in the sense that they manipulate expressions and are never present in the expression which results. For example, note the result of the following command

> diff(a*b, b)
a

10.2.5. User functions

Table 10.7 Commands for user functions.

Command	Description
clear(name)	Clears the name of a user function
declare(name(arg1, arg2, ...))	declare function name taking dummy arguments arg1, arg2, … . Derivatives assumed to be 0
define(name(arg1, arg2, ...), obj1, obj2, ...)	declare function name taking arguments arg1, arg2, … having corresponding derivatives obj1, obj2, …

Commands for specifying and manipulating user functions are listed in Table 10.7. They are used in order to define new user function, as well as the derivatives of the functions with respect to the user variables. For example, the following expression defines a function named f which takes one argument.

> define(f(x), 0.5*x)

The list after the function protoype is used to define the derivatives with respect to each of the independent variables. Once defined, the function may be used in any other expression. In additions the any expression can be used as an arguments. For example:

> diff(f(x*y),x)
((0.5 * (x * y)) * y)
> simplify((0.5 * (x * y)) * y)
(0.5 * x * (y^2))

The chain rule is applied to ensure that the derivative is correct. This can be expressed as

\[\frac{\partial}{\partial x} f \left( u, v, \ldots \right) = \frac{\partial u}{\partial x} \cdot \frac{\partial}{\partial u} f \left( u, v, \ldots \right) + \frac{\partial v}{\partial x} \cdot \frac{\partial}{\partial v} f \left( u, v, \ldots \right) + \ldots\]

The declare command is required when the derivatives of two user functions are based on one another. For example:

> declare(cos(x))
cos(x)
> define(sin(x),cos(x))
sin(x)
> define(cos(x),-sin(x))
cos(x)

When declared, a functions derivatives are set to 0, unless specified with a define command. It is now possible to use these expressions as desired.

> diff(sin(cos(x)),x)
(cos(cos(x)) * (-sin(x)))
> simplify(cos(cos(x)) * (-sin(x)))
(-cos(cos(x)) * sin(x))

10.2.6. Macro assignment

The use of macro assignment allows the substitution of expressions into new expressions. Every time a command is successfully used, the resulting expression is assigned to a special macro definition, $_.

In this example, the result of the each command is substituted into the next.

> a+b
(a + b)
> $_-b
((a + b) - b)
> simplify($_)
a

In addition to the default macro definition, it is possible to specify a variable identifier by using the $ character followed by an alphanumeric string beginning with a letter. In addition to letters and numbers, a _ character may be used as well. A macro which has not previously assigned will implicitly use 0 as its value.

This example demonstrates the use of macro assignment.

> $a1 = a + b
(a + b)
> $a2 = a - b
(a - b)
> simplify($a1+$a2)
(2 * a)

10.3. Invoking SYMDIFF from DEVSIM

10.3.1. Equation parser

The devsim.symdiff() should be used when defining new functions to the parser. Since you do not specify regions or interfaces, it considers all strings as being independent variables, as opposed to models. Model Commands presents commands which have the concepts of models. A ; should be used to separate each statement.

This is a sample invocation from DEVSIM

% symdiff(expr="subst(dog * cat, dog, bear)")
(bear * cat)

10.3.2. Evaluating external math

The devsim.register_function() is used to evaluate functions declared or defined within SYMDIFF. A Python procedure may then be used taking the same number of arguments. For example:

from math import cos
from math import sin
symdiff(expr="declare(sin(x))")
symdiff(expr="define(cos(x), -sin(x))")
symdiff(expr="define(sin(x),  cos(x))")
register_function(name="cos", nargs=1)
register_function(name="sin", nargs=1)

The cos and sin function may then be used for model evaluation. For improved efficiency, it is possible to create procedures written in C or C++ and load them into Python.

10.3.3. Models

When used withing the model commands discussed in Model Commands, DEVSIM has been extended to recognize model names in the expressions. In this situation, the derivative of a model named, model, with respect to another model, variable, is then model:variable.

During the element assembly process, DEVSIM evaluates all models of an equation together. While the expressions in models and their derivatives are independent, the software uses a caching scheme to ensure that redundant calculations are not performed. It is recommended, however, that users developing their own models investigate creating intermediate models in order to improve their understanding of the equations that they wish to be assembled.