Look at the recent HZ polynomial screen, and feel as GSH said, polynomial kinda fun.
Polynomial
Junior high school is almost polynomial polynomial.
A to \ (X \) is the variable of the polynomials in an algebraic field (F. \) \ On, the function \ (A (x) \) represented in the form and:
\ (A (X) = \ SUM \ limits_ { i = 0} ^ {n- 1} a_ix ^ i \)
Common junior high school \ (5x ^ 2 + 8x + 1,10x ^ 3 + 8x ^ 2-2 \) is a polynomial.
\ (a_0, a_1, ..., a_ {n-1} \) referred to as coefficients of the polynomial.
Wherein the number of the field F may be a complex field, the real domain ...
The number and frequency of community
And the same number of junior high definition in mind the maximum number of polynomials for the entire polynomials.
If the polynomial \ (A \) of the highest-degree coefficient is nonzero \ (a_k \) , then the degree of the polynomial is \ (k \)
denoted \ (degree (A) = k \)
For example:
\ (A (X) = 8x + lOx. 3 ^ 2-2 ^ \)
\ (Degree (A) =. 3 \)
The number is bound upper bound on the number, equivalent to the number of times a given range.
Any integer greater than the degree of the polynomial can be used as the number of community polynomial.
That you really do not need the card number on the upper bound, the number may be less than the number of circles.
Polynomial as above may be any of a sector greater than or equal \ (4 \) integer.
Polynomial addition
Polynomial addition and junior high school, as a direct merger of similar items on it.
Bound to two times \ (n-\) polynomial added polynomial boundary also \ (n-\)
\(A(x)=\sum\limits_{i=0}^{n-1}a_ix^i\)
\(B(x)=\sum\limits_{i=0}^{n-1}b_ix^i\)
\(C(x)=A(x)+B(x)\)
\(C(x)=\sum\limits_{i=0}^{n-1}(a_i+b_i)x^i\)
Note: The number of different circles can certainly increase can take, the equivalent of high-order coefficient is 0.
Polynomial multiplication
Polynomial multiplication and junior high school, as are a direct ride into a re-merger of similar items on it.
Bound to two times \ (n-\) polynomial multiplying it is bound is a number \ (2n-1 \) polynomial, since it is assumed the number of polynomials is two \ (. 1-n-\) , can take out The maximum number of \ (2N-2 \) , the number of bound \ (. 1-2N \) .
Polynomial multiplication can be seen is a convolution.
\(A(x)=\sum\limits_{i=0}^{n-1}a_ix^i\)
\(B(x)=\sum\limits_{i=0}^{n-1}b_ix^i\)
\(C(x)=A(x)B(x)\)
\(C(x)=\sum\limits_{i=0}^{2n-2}x_i\sum\limits_{k=0}^ia_kb_{i-k}\)
此时
\(degree(C)=degree(A)+degree(B)\)
The coefficients of the polynomial expression
The coefficients of the polynomial written as a \ (n-\) number of sustain vector.
\ ((a_0, a_1, a_2 , ..., a_ {n-1}) \)
At this point the polynomial multiplication is a convolution.
Convolution is commutative, associative law, distributive law.
This is clear from the junior high school mathematics or else establish a system to collapsed.
Point value polynomial expression
Taken lightly \ (n-\) ( \ (n-\) is the number of boundary) different \ (X \) , denoted as \ (x_i \) , \ (y_i = A (x_i) \)
thus forming a \ (n- \) points.
If the same is to take two polynomials (x_i \) \ point value expression, this multiplication of two polynomials \ (O (n) \) a.
Apparently only need to \ (y_i \) ride up on it.
Of course, in order to determine the ride out to take the polynomial \ (2n-1 \) key.
The only polynomial interpolation theorem
\ (n-\) point value points to determine the expression of a number of uniquely bound is \ (n-\) polynomial.
Obviously?
To bring in so many points you can get a set of n-ary n equation.
It is solvable out.
evaluate
Is some number to the value of the polynomial inside find it.
That is the coefficient expressed as a point value conversion expression.
Violence is \ (O (n ^ 2) \) of
Interpolation
The point value of the expression change back coefficient expression.
Violence also \ (O (n ^ 2) \) is.
Lagrange Interpolation
plural
I do not know why anyone would want to come out this strange thing.
Entrance examination should learn.
Complex can be expressed as \ (a + bi \)
where \ (I \) is the imaginary unit.
\ (i ^ 2 = -1, \ sqrt {-1} = i \)
Can be drawn in a coordinate system, it is like vector.
However, operation is slightly different and vector
Complex additions
Direct plus, merger of similar items.
Complex subtraction
Ditto
Complex multiplication
Direct take, then merge.
\ [(A + BI) (C + DI) = AC + ADI + CBI BDI ^ 2 + \]
\ [= (AC-BD) + (AD + CB) I \]
Multiply and divide complex and real numbers
Direct ride would be finished.
Complex trigonometric identities
Because the complex is painted on the inside of the axis.
It can be expressed in trigonometric functions.
Provided a plurality of die length \ (R & lt \) , and the complex and \ (X \) angle axis is \ (\ Alpha \) then the complex can be represented as \ (R (\ cos \ alpha + \ sin \ alpha i) \)
Complex multiplication is represented in a coordinate system
A module length to the square of the complex 1 as an example.
\ ((\ cos \ alpha +
\ sin \ alpha i) ^ 2 = \ cos ^ 2 \ alpha + 2 \ sin \ alpha \ cos \ alpha i- \ sin ^ 2 \ alpha \) followed by a number of trigonometric formulas.
\ (\ cos ^ 2 \ alpha
+ 2 \ sin \ alpha \ cos \ alpha i- \ sin ^ 2 \ alpha = \ cos2 \ alpha + \ sin2 \ alpha \) found that it happens to be a double angle.
As another general, this situation can be pushed to the right in all cases.
\ ((\ COS \ Alpha + \ SiN \ Alpha I) (\ COS \ Theta + \ SiN \ Theta I) \)
\ (= \ COS \ Alpha \ COS \ Theta + \ SiN \ Alpha \ COS \ Theta I + \ COS \ Alpha \ SiN \ Theta I- \ SiN \ Alpha \ SiN \ Theta \)
\ (= \ COS (\ Alpha + \ Theta) + \ SiN (\ Alpha + \ Theta) I \)
Therefore, we can get a conclusion, complex multiplication is equivalent to rotating the coordinate axes.