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Figures 1 218 and 1219 show the routines used to implement the postfix machine The getTop routine returns and removes the top item in the postfix stack The binaryop routine applies topop (which is expected to be the top item in the operator stack) to the top two items on the postfix stack and replaces them with the result It also pops the operator stack (at line 33) signifying that processing for topop is complete The pow routine is presumed to exist for NumericType objects; we can either use the math library routine or adapt the one previously shown in Figure 814 Figure 1220 declares a precedence table, which stores the operator precedences and is used to decide what is removed from the operator stack The operators are listed in the same order as the enumeration type TokenType Because enumeration types are assigned consecutive indices beginning with zero, they can be used to index an array (The array initialization syntax used here was described in Section 126) We want to assign a number to each level of precedence The higher the number, the higher is the precedence We could assign the additive operators precedence 1, multiplicative operators precedence 3, exponentiation precedence 5, and parentheses precedence 99 However, we also need to take into account associativity To do so, we assign each operator a number that represents its precedence when it is an input symbol and a second number that represents its precedence when it is on the operator stack A left-associative operator has the operator stack precedence set at 1 higher than the input symbol precedence, and a right-associative operator goes the other way Thus the precedence of the + operator on the stack is 2

This simple calculation thus leads to the dramatic prediction (11.6) As a(e-e+ --+ p.-p.+) is well known (see Fig. 6.6), a measurement of the total e- e+ annihilation cross section into hadrons therefore directly counts the number of quarks, their flavors, as well as their colors. We have

1,2, ... ,27.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2 1 22 23 24 25 26 27 28 29 30 31 32 33 34

i / Process an operator by taking two items off the postfix / / stack, applying the operator, and pushing the result / / Print error if missing closing parenthesis or division by 0

if ( topop == OPAREN ) i cerr << "Unbalanced parentheses" << endl; opStack pop-back ( ) ; return;

3[(1)2 +(1)2 +(t)2]

NumericType rhs = getTop( NumericType lhs = getTop(

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The discussion in Section 3.3 extends to multiple regression. The least-squares estimates of f3 o' f31' f3z, and f33 are defined to be the values of ffio, ffi" ffi2' and ffi3 which give the2 least sum of squares of, . . , . . . , . . which give the the residuals, that is, " least value of Lei' where e i = Yi - (f3o + f3l x il + f3zx iz + f33Xi3).

); );

2 + 30)2

if ( topop == EXP ) postFixStackpush-back( pow( lhs, rhs else if ( topop == PLUS ) postFixStackpush-back( lhs + rhs ) ; else if ( topop = = MINUS ) postFixStackpush-back( lhs - rhs ) ; else if ( topop = = MULT ) postFixStackpush-back( lhs * rhs ) ; else if( topop = = DIV ) if( rhs ! = 0 ) postFixStackpush-back( lhs / rhs else

cerr < < "Division by zero" << endl; postFixStackpush-back( lhs ) ;

opStackpop-back(

foru,d,s, foru,d,s,c, for u, d, s, c, b. (11.7)

Algebraic formulas for these four least-squares estimates are very messy if they are written out for each individual parameter. But if we introduce matrix notation the formula for the vector of estimates is quite compact.

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