CS 163 Data Structures Chapter 10 Symbolic Differentiation Herbert G. Mayer, PSU Status 6/11/2015

Syllabus Problem Assessment Rules of First Derivative Data Structures and Types Make Node Copy Tree Print Tree Build Tree Simplify Main References

Problem Assessment Goal of this assignment is to design, code, and execute symbolic differentiation of mathematical formulae or equations These formulae are simple, meaning: No partial differentiation Differentiation only w.r.t a single variable And that variable is pre-defined to be ‘x’ All numeric constants are integer, and of small value, i.e. single decimal digits; but leave “room” for adding up larger integer values Goal was not to focus on scanning, i.e. not on lexical analysis All variable names different from ‘x’ are constant w.r.t. differentiation toward ‘x’ If two derivatives are needed, then the output of the first can serve as input to the second differentiation So all formulae are simple, yet the mathematical problem remains general and highly interesting

Problem Assessment When applying the rules of differentiation, the result can be a formula with redundancies, e.g. +0, *1, /1 etc. In such cases: advisable to simplify the result Each input formula is terminated by a special symbol, the ‘$’ character For example: Input f(x) = 2*x+x*3+5+6*x$ Normalized input f(x) = ((((2*x)+(x*3))+5)+(6*x)) $ First derivative output f’(x) = (((((0*x)+(2*1))+((1*3)+(x*0)))+0)+((0*x)+(6*1))) Simplified output: 11

Rules of First Derivative Toward x Function references u and v below are functions of x, i.e. u(x) and v(x) the ‘ operator symbolizes the first derivative f(x) f’(x) f(x) = some variable != x f’(x) = 0 f(x) = integer constant f(x) = x f’(x) = 1 f(x) = u + v f’(x) = u’ + v’ f(x) = u - v f’(x) = u’ - v’ f(x) = u * v f’(x) = u’*v + u*v’ f(x) = u / v f’(x)= (u’*v -u*v’) / v2 f(x) = ln(u) = & u f’(x) = u’ / u f(x) = u ^ v f’(x) = u' * v * u ^ ( v - 1 ) + & u * v' * u ^ v Example: f(x) = x ^ x f’(x) = 1 * x * x^(x–1) + & x * 1 * x^x = x^x + & x * x^x = (& x + 1) * x^x

Data Structures and Types Element of symbolic-differentiation is a node Each mathematical function f(x) is represented internally as a binary tree, pointed to by “root” Root’s type is pointer to structure of node type Any node is either: A literal with a stored integer value - single decimal digit for now! A variable, which could either be the select variable ‘x’ or some other An operator, stored as a single character, like ‘+’ ‘*’ ‘/’ ... Each node has all the following fields, sometimes NOT needed: Class, specifying enumeration type { Literal, Identifier, Operator } The single character Symbol, e.g. ‘x’ for the special variable The integer literal value LitVal to remember the integer value, e.g. 7 And node pointers Left and Right to the respective subtrees

Data Structures and Types // each node has 1 of these class states: // a Literal, an Identifier (for variable), or an Operator. // Parenthesized expressions have been reduced typedef enum { Literal, Identifier, Operator } NodeClass; typedef struct NodeType * NodePtr; // forward announcement // now comes the actual node-type structure, // using the forward declared pointer type: NodePtr typedef struct NodeType { NodeClass Class; // 1 of the 3 classes. char Symbol; // store: Identifier, Operator int LitVal; // if Literal, this is its value NodePtr Left; // subtree NodePtr Right; // subtree } s_node_tp;

Make Node // malloc() new node from heap. All fields are passed in; // return the pointer to the new node to caller NodePtr Make( NodeClass Class, char Symbol, int value, NodePtr Left, NodePtr Right ) { // Make NodePtr Node = (NodePtr)malloc( sizeof( struct NodeType ) ); ASSERT( ... node’s space is really there ... ); Node->Class = Class; Node->Symbol = Symbol; Node->LitVal = value; Node->Left = Left; Node->Right = Right; return Node; } //end Make

Copy Tree // recursively copy tree pointed to by Root. // return a pointer to the copy to caller NodePtr Copy( NodePtr Root ) // clever code!!! { // Copy if ( NULL == Root ) { return NULL; }else{ return Make( Root->Class, Root->Symbol, Root->LitVal, Copy( Root->Left ), Copy( Root->Right ) ); } //end if } //end Copy

Print Tree void PrintTree( NodePtr Root ) { // PrintTree if ( Root != NULL ) { if ( Root->Class == Operator ) { printf( "(" ); } //end if PrintTree( Root->Left ); if ( Root->Class == Literal ) { printf( "%d", Root->LitVal ); // prints ints > 9 }else{ printf( "%c", Root->Symbol ); PrintTree( Root->Right ); printf( ")" ); } //end PrintTree

Build Tree for Expression() Quick BNF Intro (Backus Naur Form) Grammar is a set of rules, defining a language Rules define nonterminals via terminals, nonterminals, or special symbols (like ‘+’ or ‘;’ or ‘/’) Left side nonterminal : is defined by right side One nonterminal is the start symbol, here Expression : operator separates left from right side | introduces another alternative on r.h.s. { and } group 0 or more phrases on r.h.s. Special symbols are numeric literals (e.g. 6), identifiers (e.g. y), and char and string literals (e.g. ‘#’)

Build Tree for Expression() Make C program from BNF Grammar For each left nonterminal define a suitable C function by that nonterminal name For each nonterminal used, call that C function For each terminal used in a phrase in a non-first position, require that symbol, i.e. error if not found For each terminal at the start of an alternative, see if this terminal is in the source, then enter that alternative, else find another alternative Insert needed semantic actions

Build Tree for Expression() Expression : Term { plus_op Term } // start symbol plus_op : ‘+’ | ‘-’ Term : Factor { mult_op Factor } mult_op : ‘*’ | ‘/’ Factor : Primary { ‘^’ Primary } Primary : IDENT | LITERAL | ‘(‘ Expression ‘)’ | ‘&’ Primary // for ln()

Build Tree: Expression() // parse expression and build tree // using Term() and higher priority functions/ops // all returning pointers to nodes // in Expression() handle ‘+’ and ‘-’ operators NodePtr Expression() { // Expression char Op; // remember ‘+’ or ‘-’ NodePtr Left = Term(); // handle all higher prior. while ( NextChar == ‘+’ || NextChar == ‘-’ ) { Op = NextChar; // remember ‘+’ or ‘-’ GetNextChar(); // skip Op ‘+’ or ‘-’ // note 0 below for LitVal is just a dummy Left = Make( Operator, Op, 0, Left, Term() ); } //end while return Left; } //end Expression

Build Tree: Term() // multiply operators ‘*’ and ‘/’, later add ‘%’ NodePtr Term( ) { // Term char Op; // remember ‘*’ or ‘/’ NodePtr Left = Factor(); while ( NextChar == ‘*' || NextChar == ‘/' ) { Op = NextChar; // remember ‘*’ or ‘/’ GetNextChar(); // skip over Op // note 0 below for LitVal is just a dummy Left = Make( Operator, Op, 0, Left, Factor() ); } //end while return Left; } //end Term

Build Tree: Factor() Left-Assoc. // exponentiation operator ‘^’ left-associatively NodePtr Factor() { // Factor NodePtr Left = Primary(); while ( NextChar == ‘^’ ) { GetNextChar(); // skip over ‘^’ Left = Make( Operator, ‘^’, 0, Left, Primary() ); } //end while return Left; } //end Factor // Think about left- versus right-associativity!!! // How would you change the code –-and grammar— // if indeed you make ‘^’ right associative?

Build Tree: Factor () Right-Assoc. // exponentiation operator ‘^’ right-associative NodePtr Factor() { // Factor NodePtr Left = Primary(); if ( NextChar == ‘^’ ) { GetNextChar(); // skip over ‘^’ Left = Make( Operator, ‘^’, 0, Left, Factor() ); } //end if return Left; } //end Factor // now multiple ^ operators are handled right-to-left // in line with common precedence of exponentiation

Build Tree: Primary() NodePtr Primary( ) { // Primary char Symbol = NextChar; // first_set = { ‘(‘, ‘&’, IDENT, LIT } NodePtr Temp; GetNextChar(); // skip over current Symbol if ( IsDigit( Symbol ) ) { // end node: don’t recurse return Make( Literal, Symbol, (int)(Symbol-'0’), NULL, NULL ); }else if ( IsLetter( Symbol ) ) { // also end node: don’t recurse return Make( Identifier, tolower( Symbol ), 0, NULL, NULL ); }else if ( ‘(‘ == Symbol ) { Temp = Expression(); Must_Be( ‘)’ ); return Temp; }else if ( Symbol == '&' ) { return Make( Operator, '&', 0, NULL, primary() ); }else{ printf( "Illegal character '%c'.\n", Symbol ); return NULL; } //end if // impossible to reach! No need to check Herb!! } //end Primary

Derive, 1 // Real action: Derive(Root) derives tree pointed to by Root // First left, then right subtree // When done, focus on the Root node // Both u and v are f(x) // derive( x ) = 1 -- derive x -> 1 // derive( a ) = 0 -- any variable derived except x -> 0 // derive( # ) = 0 -- any number derived -> 0 // derive( u + v ) = u' + v' -- where u = f(x) and v = g(x) // derive( u - v ) = u' - v' // derive( u * v ) = u' * v + u * v' // derive( u / v ) = u' * v - u * v' / ( v * v) // derive( u ^ v ) = u' * v * u ^ ( v - 1 ) + ln u * v' * u ^ v // derive( ln u ) = derive( & u ) = u' / u

Derive, 2 NodePtr Derive( NodePtr Root ) { // Derive if ( Root == NULL ) { return NULL; }else{ switch ( Root->Class ) { case Literal: return Make( Literal, '0', 0, NULL, NULL ); case Identifier: if ( ( Root->Symbol == 'x' ) || ( Root->Symbol == 'X' ) ) { return Make( Literal, '1', 1, NULL, NULL ); } //end if case Operator: switch ( Root->Symbol ) { case '+': case '-': return Make( Operator, Root->Symbol, 0, Derive( Root->Left ), Derive( Root->Right ) ); case '*': return Make( Operator, '+', 0, . . Next page

Derive, 3 case '*': return Make( Operator, '+', 0, Derive( Root->Left ), Copy( Root->Right ) ), Copy( Root->Left ), Derive( Root->Right ) ) ); case '/': return Make( Operator, '/', 0, Make( Operator, '-', 0, Copy( Root->Left ), Derive( Root->Right ) ) ), Copy( Root->Right ), Copy( Root->Right ) ) ); case '^': . . . Next page

Derive, 4 case '^': return Make( Operator, '+', 0, Make( Operator, '*', 0, Derive( Root->Left ), Make( Operator, '*', 0, Copy( Root->Right ), Make( Operator, '^', 0, Copy( Root->Left ), Make( Operator, '-', 0, Copy( Root->Right ), Copy( & OneNode ) ) ) ), Make( Operator, '*', 0, Make( Operator, '&', 0, NULL, Copy( Root->Left ) ), Derive( Root->Right ) ), Make( Operator, '^', 0, Copy( Root->Left ), Copy( Root->Right ) ) ); case '&': . . . Next page

Derive, 5 case '&': if ( Root->Left != NULL ) { printf( "ln has only one operand.\n" ); } //end if return Make( Operator, '/', 0, Derive( Root->Right ), Copy( Root->Right ) ); default: printf( "Impossible operator.\n" ); return NULL; } //end switch Root->Symbol printf( "Unknown Root->Class\n" ); } //end switch Root->Class

Opportunities for Simplification # Original expression Simplified expression 1 x + 0 x 2 0 + x 3 x - 0 4 x - x 5 x * 0 6 0 * x 7 x * 1 8 1 * x 9 x / x 10 x / 1 11 x ^ 1 12 x ^ 0 13 1 ^ x 14 & 1

Simplify, 1 NodePtr Simplify( NodePtr Root ) { // Simplify int val = 0; // accumulate integer values from + - * etc. if ( !Root ) { return Root; }else{ switch ( Root->Class ) { case Literal: case Identifier: case Operator: Root->Left = Simplify( Root->Left ); Root->Right = Simplify( Root->Right ); switch ( Root->Symbol ) { case '+': if ( IsLit( '0', Root->Left ) ) { return Root->Right; }else if ( IsLit( '0', Root->Right ) ) { return Root->Left; }else if ( BothLit( Root->Left, Root->Right ) ) { val = Root->Left->LitVal + Root->Right->LitVal; return Make( Literal, (char)( val + '0' ), val, NULL, NULL ); return Root; // no other simplifiction for ‘+’ } //end if . . .

Simplify, 2 case '-': if ( IsLit( '0', Root->Right ) ) { return Root->Left; }else if ( BothLit( Root->Left, Root->Right ) ) { val = Root->Left->LitVal - Root->Right->LitVal; return Make( Literal, (char)( val + '0' ), val, NULL, NULL ); }else if ( IsEqual( Root->Left, Root->Right ) ) { return & NullNode; }else{ return Root; } //end if case '*': if ( IsLit( '1', Root->Left ) ) { return Root->Right; }else if ( IsLit( '1', Root->Right ) ) { }else if ( IsLit( '0', Root->Left ) || IsLit( '0', Root->Right ) ) { }//end if case '/': if ( IsLit( '1', Root->Right ) ) { }else if ( IsLit( '0', Root->Left ) ) { return & OneNode; case '^': if ( IsLit( '0', Root->Right ) ) { // x^0 = 1 }else if ( IsLit( '1', Root->Right ) ) { // x^1 = x }else if ( IsLit( '1', Root->Left ) ) { // 1^x = 1 . . .

Two Equal Trees Students write code for: bool IsEqual( NodePtr Left, NodePtr Right )

Two Equal Trees // return true only if both subtrees left and right are equal bool IsEqual( NodePtr Left, NodePtr Right ) { // IsEqual if ( ( !Left ) && ( !Right ) ) { return TRUE; }else if ( NULL == Left ) { // Right is known to be not NULL return FALSE; }else if ( NULL == Right ) { // Left is known to be NOT NULL }else if ( ( Left->Class == Literal ) && ( Right->Class == Literal ) ) { return ( Left->LitVal ) == ( Right->LitVal ); }else if ( ( Left->Class == Identifier ) && ( Right->Class == Identifier )){ return ( Left->Symbol ) == ( Right->Symbol ); }else{ // must be Operator; same? if ( ( Left->Symbol ) == ( Right->Symbol ) ) { // IsEqual yields true, only if both subtrees are equal return ( IsEqual( Left->Left, Right->Left ) && IsEqual( Left->Right, Right->Right ) ) || ( is_associative( Left->Symbol ) && IsEqual( Left->Left, Right->Right ) && IsEqual( Left->Right, Right->Left ) ); } //end if printf( "Impossible to reach in IsEqual.\n" ); } //end IsEqual

main() int main () { // main: Differentiation NodePtr root = NULL; Initialize(); root = Expression(); VERIFY( ( NextChar == '$' ), "$ expected, not found\n" ); SHOW( " original f(x) = ", root ); root = Simplify( root ); SHOW( " Simplified f(x) = ", root ); root = Derive( root ); SHOW( " derived f'(x) = ", root ); SHOW( " reduced f'(x) = ", root ); Or else: print_tree( simplify( derive( simplify( expression( root ))))); return 0; } //end main: Differentiation

References Differentiation rules, implementation code samples: http://www.codeproject.com/KB/recipes/Differentiation.aspx More code samples in Lisp: http://mitpress.mit.edu/sicp/full- text/sicp/book/node39.html