6.6.2 Kinds of Declarations

The type-specifier part of a declaration (that is, the second prompt in the s d command) can be a type symbol, an interval, or a vector consisting of zero or more type symbols followed by zero or more intervals or numbers that represent the set of possible values for the variable.

[ [ a, [1, 2, 3, 4, 5] ]
  [ b, [1 .. 5]        ]
  [ c, [int, 1 .. 5]   ] ]

Here a is declared to contain one of the five integers shown; b is any number in the interval from 1 to 5 (any real number since we haven’t specified), and c is any integer in that interval. Thus the declarations for a and c are nearly equivalent (see below).

The type-specifier can be the empty vector ‘[]’ to say that nothing is known about a given variable’s value. This is the same as not declaring the variable at all except that it overrides any All declaration which would otherwise apply.

The initial value of Decls is the empty vector ‘[]’. If Decls has no stored value or if the value stored in it is not valid, it is ignored and there are no declarations as far as Calc is concerned. (The s d command will replace such a malformed value with a fresh empty matrix, ‘[]’, before recording the new declaration.) Unrecognized type symbols are ignored.

The following type symbols describe what sorts of numbers will be stored in a variable:

int

Integers.

numint

Numerical integers. (Integers or integer-valued floats.)

frac

Fractions. (Rational numbers which are not integers.)

rat

Rational numbers. (Either integers or fractions.)

float

Floating-point numbers.

real

Real numbers. (Integers, fractions, or floats. Actually, intervals and error forms with real components also count as reals here.)

pos

Positive real numbers. (Strictly greater than zero.)

nonneg

Nonnegative real numbers. (Greater than or equal to zero.)

number

Numbers. (Real or complex.)

Calc uses this information to determine when certain simplifications of formulas are safe. For example, ‘(x^y)^z’ cannot be simplified to ‘x^(y z)’ in general; for example, ‘((-3)^2)^1:2’ is 3, but ‘(-3)^(2*1:2) = (-3)^1’ is -3. However, this simplification is safe if z is known to be an integer, or if x is known to be a nonnegative real number. If you have given declarations that allow Calc to deduce either of these facts, Calc will perform this simplification of the formula.

Calc can apply a certain amount of logic when using declarations. For example, ‘(x^y)^(2n+1)’ will be simplified if n has been declared int; Calc knows that an integer times an integer, plus an integer, must always be an integer. (In fact, Calc would simplify ‘(-x)^(2n+1)’ to ‘-(x^(2n+1))’ since it is able to determine that ‘2n+1’ must be an odd integer.)

Similarly, ‘(abs(x)^y)^z’ will be simplified to ‘abs(x)^(y z)’ because Calc knows that the abs function always returns a nonnegative real. If you had a myabs function that also had this property, you could get Calc to recognize it by adding the row ‘[myabs(), nonneg]’ to the Decls matrix.

One instance of this simplification is ‘sqrt(x^2)’ (since the sqrt function is effectively a one-half power). Normally Calc leaves this formula alone. After the command s d x RET real RET, however, it can simplify the formula to ‘abs(x)’. And after s d x RET nonneg RET, Calc can simplify this formula all the way to ‘x’.

If there are any intervals or real numbers in the type specifier, they comprise the set of possible values that the variable or function being declared can have. In particular, the type symbol real is effectively the same as the range ‘[-inf .. inf]’ (note that infinity is included in the range of possible values); pos is the same as ‘(0 .. inf]’, and nonneg is the same as ‘[0 .. inf]’. Saying ‘[real, [-5 .. 5]]’ is redundant because the fact that the variable is real can be deduced just from the interval, but ‘[int, [-5 .. 5]]’ and ‘[rat, [-5 .. 5]]’ are useful combinations.

Note that the vector of intervals or numbers is in the same format used by Calc’s set-manipulation commands. See Set Operations using Vectors.

The type specifier ‘[1, 2, 3]’ is equivalent to ‘[numint, 1, 2, 3]’, not to ‘[int, 1, 2, 3]’. In other words, the range of possible values means only that the variable’s value must be numerically equal to a number in that range, but not that it must be equal in type as well. Calc’s set operations act the same way; ‘in(2, [1., 2., 3.])’ and ‘in(1.5, [1:2, 3:2, 5:2])’ both report “true.”

If you use a conflicting combination of type specifiers, the results are unpredictable. An example is ‘[pos, [0 .. 5]]’, where the interval does not lie in the range described by the type symbol.

“Real” declarations mostly affect simplifications involving powers like the one described above. Another case where they are used is in the a P command which returns a list of all roots of a polynomial; if the variable has been declared real, only the real roots (if any) will be included in the list.

“Integer” declarations are used for simplifications which are valid only when certain values are integers (such as ‘(x^y)^z’ shown above).

Calc’s algebraic simplifications also make use of declarations when simplifying equations and inequalities. They will cancel x from both sides of ‘a x = b x’ only if it is sure x is non-zero, say, because it has a pos declaration. To declare specifically that x is real and non-zero, use ‘[[-inf .. 0), (0 .. inf]]’. (There is no way in the current notation to say that x is nonzero but not necessarily real.) The a e command does “unsafe” simplifications, including canceling ‘x’ from the equation when ‘x’ is not known to be nonzero.

Another set of type symbols distinguish between scalars and vectors.

scalar

The value is not a vector.

vector

The value is a vector.

matrix

The value is a matrix (a rectangular vector of vectors).

sqmatrix

The value is a square matrix.

These type symbols can be combined with the other type symbols described above; ‘[int, matrix]’ describes an object which is a matrix of integers.

Scalar/vector declarations are used to determine whether certain algebraic operations are safe. For example, ‘[a, b, c] + x’ is normally not simplified to ‘[a + x, b + x, c + x]’, but it will be if x has been declared scalar. On the other hand, multiplication is usually assumed to be commutative, but the terms in ‘x y’ will never be exchanged if both x and y are known to be vectors or matrices. (Calc currently never distinguishes between vector and matrix declarations.)

See Matrix and Scalar Modes, for a discussion of Matrix mode and Scalar mode, which are similar to declaring ‘[All, matrix]’ or ‘[All, scalar]’ but much more convenient.

One more type symbol that is recognized is used with the H a d command for taking total derivatives of a formula. See Calculus.

const

The value is a constant with respect to other variables.

Calc does not check the declarations for a variable when you store a value in it. However, storing -3.5 in a variable that has been declared pos, int, or matrix may have unexpected effects; Calc may evaluate ‘sqrt(x^2)’ to ‘3.5’ if it substitutes the value first, or to ‘-3.5’ if x was declared pos and the formula ‘sqrt(x^2)’ is simplified to ‘x’ before the value is substituted. Before using a variable for a new purpose, it is best to use s d or s D to check to make sure you don’t still have an old declaration for the variable that will conflict with its new meaning.