+ y Conjugate
The complex conjugate of the number y
+ 3 3 + 3j5 3j_5
1. Test z is real not complex
if. z=+z do. ... end.
2. Find the real part of z
z=: 3j4 -: z+ +z NB. (-:) is: Halve 3
A better solution is 9&o. (see Circle Functions (o.).
1. Complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs.
If y is real, then (+y) is the same as y
+ 7 0 _7 7 0 _7
2. J supports complex numbers and returns them as required by a calculation.
The way to write the scalar numeral having real part 3 and imaginary part 4i is: (3j4).
sqrt=: 3 : 'y^0.5' NB. (sqrt y) is y to the power of 0.5 sqrt=: ^&0.5 NB. (tacit alternative) sqrt 49 7 sqrt _1 0j1 + sqrt _1 0j_1 |sqrt _1 NB. The sq root of _1 has magnitude 1 1 | z=: 3j4 NB. vector repn of z is 3-4-5 triangle 5 NB. hence its magnitude is the hypotenuse + z 3j_4 | +z NB. Conjugate of z has same magnitude as z 5
x + y Plus
2 + 3 5
Either or both of x, y can be atoms.
x=: 5 y=: 2 3 4 x + y 7 8 9 y + x 7 8 9
1. Increment an array by the same amount throughout
100 + 0 1 2 100 101 102
2. Sum the numbers in a given list
+/0 1 2 3 6
x=: 100 200 y=: 2 3$i.6 x + y 100 101 102 203 204 205 x=: 100 200 300 x + y |length error | x +y[x=:100 200 300[y=:2 3$i.6
x +"1 y 100 201 302 103 204 305
Use These Combinations
Combinations using x + y that have exceptionally good performance include:
What It Does Type;
Syntax Primitives permitted in place of f Variants;
Count number of places where x f y is true Permitted: Boolean, integer, floating point, byte, symbol (not unicode).
x and y need not be the same precision.
x ([: +/ f) y x +/@:f y = ~: < <: > >: e. E. Permitted: (f!:0) (parentheses obligatory!) to force exact comparison.
J recognizes FLO only if f returns an atom or list.
Avoids computing entire x f y
Bug warning: if f is e. it does (,@e.) rather than e. regardless of ranks of arguments
What it does Type;
Count number of cells of y that match m-items +/@e.&m y Bug warning: it does (,@e.) rather than e. Reductions on infixes Boolean, integer, floating-point x +/\ y <. >. in place of + much faster than alternatives Mean on infixes integer and floating-point x (+/%#)\ y x positive
*. = ~: in place of +
much faster than alternatives Boolean reductions on partitions Boolean x +//. y = <. >. +. * *. ~: in place of + avoids building argument cells Reductions on partitions integer, floating-point x +//. y <. >. in place of + avoids building argument cells Find mean of each partition x (+/ % #)/. y avoids building argument cells Polynomial Multiplication x +//.@(*/) y avoids building argument cells Polynomial Multiplication (Boolean) Boolean x ~://.@(*./) y
x ~://.@(+./) y
x +//.@(*./) y
x +//.@(+./) y
avoids building argument cells Sum along diagonals +//. y avoids building argument cells Mean with rank (+/ % #) y Supported as a primitive by (+/ % #)"n