transcendental |ˌtransenˈdentl| |ˌˈtrønˈ(t)sɛnˌdɛn(t)l| |
ˌˈtrøn(t)snˌdɛn(t)l| |ˈtransɛnˌdɛnt(ə)l| |ˈtrɑːn-|
adjective
1 of or relating to a spiritual or nonphysical realm : the
transcendental importance of each person's soul.
• (in Kantian philosophy) presupposed in and necessary to
experience; a priori.
• relating to or denoting Transcendentalism.
2 Mathematics (of a number, e.g., e or π) real but not a root of an
algebraic equation with rational roots.
• (of a function) not capable of being produced by the algebraical
operations of addition, multiplication, and involution, or the
inverse operations.
Since transcendental functions cannot be produced by algebraical
operations, for those that care, algorithms should correct terms
which can be determined by other means.
Donna
[EMAIL PROTECTED]
On 5-Dec-06, at 5:37 PM, Roger Hui wrote:
Discussions subsequent to our interchange indicated
how one can compute sin to any finite number of
digits. In any case, the power series for sin and
cos are topics treated in a first-year calculus
course. x=: 3.141592653589793 is slightly less than
true pi, that is, x=pi-c where c is a number with
small magnitude. By the sum formula
(sin a+b)=((sin a)*cos b)+(sin b)*cos a
and so (sin pi-c)=((sin pi)*cos -c)+(sin -c)*cos pi.
The first term simplifies to 0 since sin pi is 0.
The second term simplifies to sin c since sin -c is
- sin c and cos pi is _1 . Thus:
x=: 3141592653589793x % 10^15x
pi=: ((10^40x) %~ <[EMAIL PROTECTED] 10^40x)
c=: pi - x
0j_8 ": c
2.38462643e_16
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