8E. Complex Numbers
m0=: cnj=: + |
Conjugate |
m1=: mag=: | |
Magnitude |
m2=: %:@(cnj*]) |
" |
m3=: rai=: +. |
Real and imaginary parts |
m4=: maa=: *. |
Magnitude and angle |
m5=: irai=: rai^:_1 |
Inverse rai |
m6=: imaa=: maa^:_1 |
Inverse maa |
m7=: rou=: [:^ 0j2p1&% * i. |
Roots of unity |
m8=: rpg=: rai@rou |
Regular polygon |
d9=: zero=: ] * 10&^@-@[ < | |
Zero any real y less than 10^-x in mag |
m10=: z=:({.,{:*1e_6"_<%~/@:|)&.rai |
Zero imaginary if relatively small |
m11=: (1e_10&$:) : (j./"1@((] * (<:|)) +.)) |
Clean y |
The function z may be used to zero any imaginary part that is relatively small compared to the corresponding real part. For example:
(] ,: z) a=:3+j.10^-2*i. 5 3j1 3j0.01 3j0.0001 3j1e_6 3j1e_8 3j1 3j0.01 3j0.0001 3 3 z a 3j1 3j0.01 3j0.0001 3 3
Complex numbers can be scaled by multiplication by a real number, and shifted and rotated by addition and multiplication by complex numbers. For example:
]a=: rou 3 NB. Third roots of unity 1 _0.5j0.866025 _0.5j_0.866025 |a NB. Lie on the unit circle 1 1 1 1ad30 NB. Complex of mag 1 and angle of 30 degrees 0.866025j0.5 6&zero@rai&.> (] ; 3j4&+ ; 1ad30&* ; 2ad60&*) a +---------------------------------------------------+ | 1 0| 4 4| 0.866025 0.5| 1 1.73205| | 0.5 0.866025|2.5 4.86603|_0.866025 0.5|_2 0| |_0.5 0.866025|2.5 3.13397| 0 _1| 1 _1.73205| +---------------------------------------------------+ Coordinates Shift by 3,4 Rotate by Rotate by of triangle 30 degrees 60 degrees