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2.1 Algebraic Operators

Operator: + augend addend

Addition of scalar quantities or componentwise addition of bunches is accomplished by means of the infix operator +. For example,

e2 : a:[[1, 3, 5], [2, 4, 7]];

    [1  3  5]
e2: [       ]
    [2  4  7]

e3 : b:[2, 4];

e3: [2, 4]

e4 : a + b;

    [3  5  7 ]
e4: [        ]
    [6  8  11]

e5 : 3 + 2;

e5: 5

e6 : c + b;

e6: [2 + c, 4 + c]

e7 : e1 + e5;

         2   2
e7: 5 + (8 a + 12 a ) b
Operator: - minuend subtrahend
Operator: - subtrahend

The symbol - is used to denote either the binary infix operator subtraction or the unary minus.

e1 : -[1,2,3];

e1: [-1, -2, -3]

e2 : 3-7;

e2: -4
Operator: +/- minuend subtrahend
Operator: -/+ minuend subtrahend
Operator: +/- augend
Operator: -/+ augend

Jacal allows the use of +/- and -/+ as ambiguous signs (unary plus-or-minus, unary minus-or-plus) and as ambiguous infix operators (binary plus-or-minus, binary minus-or-plus). The value +/- is also represented by the constant %sqrt1, while -/+ is represented by -%sqrt1.

e7 : u:+/-3;

e7: 3 %sqrt1

e8 : u^2;

e8: 9

e9 : +/-(u);

e9: 3

e10 : u-/+3;

e10: b-/+(3 %sqrt1, 3)
Operator: * multiplicand1 multiplicand2

Multiplication of scalar expressions such as numbers, polynomials, rational functions and algebraic functions is denoted by the infix operator *. For example,

e1 : (2 + 3 * a) * 4 * a * b^2;

               2   2
e1: (8 a + 12 a ) b

One can also use * as an infix operator on bunches. In that case, it operates componentwise, in an appropriate sense. If the bunches are square matrices, the operator * multiplies corresponding entries of the two factors. It does not perform matrix multiplication. To multiply matrices one instead uses the operator . (i.e., a period). More generally, any binary scalar operator other than ^ can be used on bunches and acts componentwise.

Operator: / dividend divisor

The symbol for division in Jacal is /. For example, the value returned by 6 / 2 is 3.

e3 : (x^2 - y^2) / (x - y);

e3: x + y
Operator: ^ expression exponent

The infix operator ^ is used for exponentiation of scalar quantitites or for componentwise exponentiation of bunches. For example, 2^5 returns 32. Unlike the other scalar infix operators, one cannot use ^ for component-wise operations on bunches. Furthermore, one should not try to use ^ to raise a square matrix to a power. Instead, one should use ^^.

e7 : (1+x)^4;

                 2      3    4
e7: 1 + 4 x + 6 x  + 4 x  + x
Operator: = expression1 expression2

In Jacal, the equals sign = is not used for conditionals and it is not used for assignments. To assign one value to another, use either : or :=. The operator = merely returns a value of the form 0 = expression. The value returned by a = b, for example is 0 = a - b.

e6 : 1=2;

e6: 0 = -1
Operator: || Z1 Z2

The infix operator || is from electrical engineering and represents the effective impedance of the parallel connection of components of impedances Z1 and Z2:

e1 : Z1 || Z2;

     Z1 Z2
e1: -------
    Z1 + Z2

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