MATLAB課堂作業: 作業八

1. Find the multiplication result of two polynorminals, in which p=133x^5+122x^3+1, q=2x^4+100x^2+1.
ANS:
>> p=[133 0 122 0 0 1];
>> q=[2 0 100 0 1];
>> M=conv(p,q)

M =

266 0 13544 0 12333 2 122 100 0 1
故兩多項式乘積M= 266x⁹+13544x⁷+12333x⁵+2x⁴+122x³+100x²+1

2. A polynorminal is defined as f=100x^3+23x^2+x+45. Find the value f(x) if x is a magic matrix in the order of 5.
ANS:
>> x=magic(5);
>> f=[100 23 1 45];
>> v=polyval(f,x)

v =

      498009     1395717         169       52725      342735
     1228935       13125       35479      278967      415549
        6817       22479      223645      809265     1075999
      102355      176169      694267      936309        2955
      135939      590715     1576945         939       74817

3. Using p and q defined in the item 1, find the quotient(商數) and residue(餘數) of p/q.
ANS:
>> p=[133 0 122 0 0 1];
>> q=[2 0 100 0 1];
>> [s,r]=deconv(p,q) % s代表商數，r代表餘數

s =

   66.5000         0


r =

  1.0e+003 *

         0         0   -6.5280         0   -0.0665    0.0010

商數為66.5x，餘數為-6528x³-66.5x+1

4. Find the roots of p=0 and q=0, in which both p & q are defined in item 1.
ANS:
>> p=[133 0 122 0 0 1];
>> q=[2 0 100 0 1];
>> roots(p)

ans =

  -0.0045 + 0.9578i
  -0.0045 - 0.9578i
   0.1039 + 0.1744i
   0.1039 - 0.1744i
  -0.1988          

>> roots(q)

ans =

        0 + 7.0704i
        0 - 7.0704i
        0 + 0.1000i
        0 - 0.1000i

p共有5個根，q共有4個根

5. Fit a polynorminal curve to following data to an order of 3. x=[1:10]; y=[1210, 1866, 2301, 2564, 2724, 2881, 2879, 2915, 3010].
ANS:
>> x=[1:10];y=[1210, 1866, 2301, 2564, 2724, 2881, 2879, 2915, 3010];
>> polyfit(x,y,3)
??? Error using ==> polyfit
X and Y vectors must be the same size.
原本題目中y陣列中的元素值只有9個，與x陣列元素個數不符，因此執行指令時polyfit會有錯誤指令出現
若再加入一個自訂數值3178，則可算出最符合的多項式：

>> x=[1:10];y=[1210,1866,2301,2564,2724,2881,2879,2915,3010,3178];
>> polyfit(x,y,3)

ans =

  1.0e+003 *

    0.0065   -0.1372    1.0049    0.3410

多項式為6.5x³-137.2x²+100.49x+341