宣言の型式記述部(s d コマンドの2番目のプロンプト)は、 1個の型式記号か1個の区間、あるいはそれらのベクトルです。 ベクトルの場合は、その要素を次の順に並べます。
[ [ a, [1, 2, 3, 4, 5] ] [ b, [1 .. 5] ] [ c, [int, 1 .. 5] ] ]
a には、
示された5つの整数のどれかひとつの値がストアされる、と宣言されています。
b は閉区間1〜5の範囲の(型は指定していないので実数)値です。
c は示された区間内の整数です。
従って a と c の宣言はほとんど同じです(後述)。
型式記述部には、空ベクトル `[]' も指定できて、
その場合は「指定の変数に関する情報無し」という意味になります。
これはその変数について何も指定しないのと全く同じですが、
他に All 宣言がある場合にはこうしておかないと適用されてしまいます。
Decls の初期値は空ベクトル `[]' です。
Decls に何もストアされていなかったり、
そこにストアされている値が無効だったりすると、
それは無視されて、宣言は何も無いとみなされます。
(s d コマンドは、新しい宣言を書込む前に、
そのような壊れた値を初期値の空行列 `[]' に置換えます。)
認識できない型式記号は無視されます。
ある変数にどんなデータ型式がストアされるかを宣言する型式記号を、 以下にリストします。
int
numint
frac
rat
float
real
pos
nonneg
number
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. Set Operations using Vectors 参照 .
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).
Another command that makes use of declarations is a s, when
simplifying equations and inequalities. It 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 cancelling `x' from the equation when `x' is
not known to be nonzero.
Another set of type symbols distinguish between scalars and vectors.
scalar
vector
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.)
行列モードとスカラ・モード 参照 , 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. 微積分(Calculus) 参照 .
const
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.
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