# Arithmetic¶

### Sum a vector¶

q)x:1 2 3 4 5
q)sum x
15


### Sum columns of matrix¶

q)x:1+3 4#til 12
q)x
1 2  3  4
5 6  7  8
9 10 11 12
q)sum x
15 18 21 24


### Sum rows of matrix¶

q)x:1+3 4#til 12
q)x
1 2  3  4
5 6  7  8
9 10 11 12
q)sum each x
10 26 42


### Sum items grouped by y¶

q)show x:log 1+til 5
0 0.6931472 1.098612 1.386294 1.609438
q)y:1 4 1 4 1
q)show a:value group y
0 2 4
1 3
q)sum each x[a]
2.70805 2.079442
q)sum x[0 2 4]
2.70805
q)sum x[1 3]
2.079442
q)sum each x value group y
2.70805 2.079442


### Add vector to each row of matrix¶

q)x:1+til 4
q)show y:3 4#2+til 12
2  3  4  5
6  7  8  9
10 11 12 13
q)x+/:y
3  5  7  9
7  9  11 13
11 13 15 17


### Add vector to each column of matrix¶

q)x:1+til 2
q)show y:2 5#3+til 10
3 4 5  6  7
8 9 10 11 12
q)x+'y
4  5  6  7  8
10 11 12 13 14


## Add x to last item of y¶

q)x:100
q)y:1 2 3 4 5
q)@[y;-1+count y;+;x]
1 2 3 4 105


Or in place:

q)@[y;-1+count y;+;x]
y
q)y
1 2 3 4 105


Or.

q)y:1 2 3 4 5
q)y[count[y]-1]+:x
q)y
1 2 3 4 105


### Sum reciprocal series¶

q)x:10 9 10 7 8
q)y:80 63 70 63 64
q)sum y%x
39f


### Sum over subsets of x specified by y¶

q)show x:float$1+3 4#til 12 1 2 3 4 5 6 7 8 9 10 11 12 q)show y:float$4 3#1 0
1 0 1
0 1 0
1 0 1
0 1 0
q)x mmu y
4  6  4
12 14 12
20 22 20


### Sum squares¶

q)x:1 2 3 4 5
q)sum x*x
55


### Alternating sum¶

q)x:1+til 10
q)x
1 2 3 4 5 6 7 8 9 10
q)a:((count x)#1 -1)
q)b:x*a
q)sum b
-5
q)sum x*(count x)#1 -1
-5


### Alternating sum series¶

q)x:1+til 10
q)x
1 2 3 4 5 6 7 8 9 10
q)a:((count x)#1 -1)
q)a
1 -1 1 -1 1 -1 1 -1 1 -1
q)b:x*a
q)b
1 -2 3 -4 5 -6 7 -8 9 -10
q)sums b
1 -1 2 -2 3 -3 4 -4 5 -5
q)as:{sums x*(count x)#1 -1}
q)as[x]
1 -1 2 -2 3 -3 4 -4 5 -5


### Increase absolute value without sign change¶

q)x:0 -1 2 -3 4 -5
q)y:10
q)x+y*(x=0)+signum x
10 -11 12 -13 14 -15
q)x+y*@[signum x;where x=0;:;1]
10 -11 12 -13 14 -15


## Subtract¶

### Vector and its negative¶

q)x:1 -3 5
q)x,'neg x
1  -1
-3 3
5  -5

q)1 -1*/:x
1  -1
-3 3
5  -5


## Numeric array and its negative¶

q)show x:3+3 4#til 12
3  4  5  6
7  8  9  10
11 12 13 14
q)x,''neg x
3 -3   4 -4   5 -5   6 -6
7  -7  8  -8  9  -9  10 -10
11 -11 12 -12 13 -13 14 -14
q)((1 -1*)'')x
3 -3   4 -4   5 -5   6 -6
7  -7  8  -8  9  -9  10 -10
11 -11 12 -12 13 -13 14 -14


### First difference¶

q)show x:sums 1 2 3 4 5
1 3 6 10 15
q)deltas x
1 2 3 4 5


### Pairwise difference¶

q)x:9 3 5 2 0
q)deltas x
9 -6 2 -3 -2
q)1_ deltas x
-6 2 -3 -2
q)1_ neg deltas x
6 -2 3 2


## Multiply¶

### Zeroing an array¶

q)show x:3 4#til 12
0 1 2  3
4 5 6  7
8 9 10 11
q)x*0
0 0 0 0
0 0 0 0
0 0 0 0
q)x[;]:0
q)x
0 0 0 0
0 0 0 0
0 0 0 0


### Product¶

q)x:1 2 3 4 5
q)prd x
120


### Multiplication table of order x¶

q)mt:{{x*\:x}1+til x}
q)mt 5
1 2  3  4  5
2 4  6  8  10
3 6  9  12 15
4 8  12 16 20
5 10 15 20 25


### Square x retaining sign¶

q)x:0 -1 2 -3 4
q)x*abs x
0 -1 4 -9 16


### Multiply each column be vector¶

q)show x:2 5#1+til 10
1 2 3 4 5
6 7 8 9 10
q)y:10 100
q)x*y
10  20  30  40  50
600 700 800 900 1000


### Multiply each row by vector¶

q)show x:3 4#1+til 12
1 2  3  4
5 6  7  8
9 10 11 12
q)y:1 10 100 10000
q)x*\:y
1 20  300  40000
5 60  700  80000
9 100 1100 120000


### Vector (cross) product¶

q)x:2 8 5 6 3 1 7 7 10 4
q)y:6 9 1 1 6 7 1 4 1 5
q)((1 rotate x)*-1 rotate y)-(-1 rotate x)*1 rotate y
4 28 46 -27 -41 39 45 3 -19 -58


Or.

q)(-). prd each(1 neg\1 -1)rotate''2 2#(x;y)
4 28 46 -27 -41 39 45 3 -19 -58
q)(-).(*/')(1 neg\1 -1)rotate''2 2#(x;y)
4 28 46 -27 -41 39 45 3 -19 -58


### Alternating product¶

q)x:1 2 3 4 5
q)a:(count x)#1 -1
q)a
1 -1 1 -1 1
q)prd xexp[x;a]
1.875
q)xexp[x;a]
1 0.5 3 0.25 5


### Direct matrix product¶

q)x:1+3 2#til 6
q)y:1+2 4#til 8
q)flip each x*\:\:y
1 2 3 4     2 4 6 8     5  6  7  8  10 12 14 16
3 6 9  12   4 8 12 16   15 18 21 24 20 24 28 32
5 10 15 20  6 12 18 24  25 30 35 40 30 36 42 48
q)dp:{flip each x*\:\:y}


### Matrix product¶

q)show x:float$(1 2 3;4 5 6) 1 2 3 4 5 6 q)show y:float$(1 2;3 4;5 6)
1 2
3 4
5 6
q)x mmu y
22 28
49 64


### Dot product of vectors¶

q)x:1 2 3 4 5
q)y:10 20 30 40 50
q)sum x*y
550


### Product over subsets of x specified by y¶

q)show x:1+3 4#til 12
1 2  3  4
5 6  7  8
9 10 11 12
q)show y:4 3#1 0
1 0 1
0 1 0
1 0 1
0 1 0
q)x('[prd;xexp])\:y
3  8   3
35 48  35
99 120 99


## Divide¶

### Divide by 0¶

q)dz:{not[z]*y%x+z:x=0}
q)y:10 15 -20
q)x:2 0 0
q)y%x
5 0w -0w
q)dz[x;y]
5 0 -0f


### Divisors¶

q)dv:{where 0=x mod/:key 1+x}
q)x:363
q)dv x
1 3 11 33 121 363
q)x:365
q)dv x
1 5 73 365
q)dv 367
1 367
q)dv each 1 2 3 4 5 6 7 8 9 10
,1
1 2
1 3
1 2 4
1 5
1 2 3 6
1 7
1 2 4 8
1 3 9
1 2 5 10


### Greatest common divisor¶

q)x:6 9 12
q)1+til min x
1 2 3 4 5 6
q)x mod/:1+til min x
0 0 0
0 1 0
0 0 0
2 1 0
1 4 2
0 3 0
q)0=x mod/:1+til min x
111b
101b
111b
001b
000b
101b
q)min each 0=x mod/:1+til min x
101000b
q)where min each 0=x mod/:1+til min x
0 2
q)1+last where min each 0=x mod/:1+til min x
3


### Pairwise ratios¶

q)ratios x
2 5 5 2f
q)1_ratios x
5 5 2f


## Compare¶

### Maximum table¶

q)x:til 10
q)x&\:x
0 0 0 0 0 0 0 0 0 0
0 1 1 1 1 1 1 1 1 1
0 1 2 2 2 2 2 2 2 2
0 1 2 3 3 3 3 3 3 3
0 1 2 3 4 4 4 4 4 4
0 1 2 3 4 5 5 5 5 5
0 1 2 3 4 5 6 6 6 6
0 1 2 3 4 5 6 7 7 7
0 1 2 3 4 5 6 7 8 8
0 1 2 3 4 5 6 7 8 9


### Minimum table¶

q)x:til 10
q)x|\:x
0 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9
2 2 2 3 4 5 6 7 8 9
3 3 3 3 4 5 6 7 8 9
4 4 4 4 4 5 6 7 8 9
5 5 5 5 5 5 6 7 8 9
6 6 6 6 6 6 6 7 8 9
7 7 7 7 7 7 7 7 8 9
8 8 8 8 8 8 8 8 8 9
9 9 9 9 9 9 9 9 9 9


## Numbers¶

### Consecutive integers from x to y¶

q)x:5
q)y:10
q)x+til 1+y-x
5 6 7 8 9 10


Or.

q)x_til y+1
5 6 7 8 9 10


### Arithmetic progression of y numbers from x with step g¶

q)x:5
q)y:8
q)g:100
q)x+g*til y
5 105 205 305 405 505 605 705


### Arithmetic progression from x to y with step g¶

q)ap:{[x;y;g]x+g*til 1+ floor (y-x)%g}
q)ap[3;20;5]
3 8 13 18
q)ap[3;-20;-5]
3 -2 -7 -12 -17


### Primes to n¶

q)n:10
q)show x:1+til n
1 2 3 4 5 6 7 8 9 10
q)x mod/:x:1+til n
0 0 0 0 0 0 0 0 0 0
1 0 1 0 1 0 1 0 1 0
1 2 0 1 2 0 1 2 0 1
1 2 3 0 1 2 3 0 1 2
1 2 3 4 0 1 2 3 4 0
1 2 3 4 5 0 1 2 3 4
1 2 3 4 5 6 0 1 2 3
1 2 3 4 5 6 7 0 1 2
1 2 3 4 5 6 7 8 0 1
1 2 3 4 5 6 7 8 9 0
q)sum 0=x mod/:x:1+til n
1 2 2 3 2 4 2 4 3 4
q)2=sum 0=x mod/:x:1+til n
0110101000b
q)0b,2=sum 0=x mod/:x:1+til n
00110101000b
q)where 0b,2=sum 0=x mod/:x:1+til n
2 3 5 7
q)pn:{[n]where 0b,2=sum 0=x mod/:x:1+til n}
q)pn 30
2 3 5 7 11 13 17 19 23 29
q)/Classic:
q)p:{a where not a in distinct raze a*/:\:a:2_ til x}
q)p 10
2 3 5 7
q)p 30
2 3 5 7 11 13 17 19 23 29


### X first triangular numbers¶

q)x:6
q)sums til x
0 1 3 6 10 15


### First 10 figurate numbers of order x¶

q)fg:{x+\/10#1}
q)fg 0
1 1 1 1 1 1 1 1 1 1
q)fg 1
1 2 3 4 5 6 7 8 9 10
q)fg 2
1 3 6 10 15 21 28 36 45 55
q)fg 3
1 4 10 20 35 56 84 120 165 220
q)fg 4
1 5 15 35 70 126 210 330 495 715


## Parts of numbers¶

### Fractional part¶

q)x:0 1 -2 3.4 -5.6 -6.1
q)x mod 1
0 0 0 0.4 0.4 0.9


### Integer and fractional parts of positive x¶

q)x:12.3 23.4 5.33 8.999
q){f,'x-f:floor x}x
12 0.3
23 0.4
5  0.33
8  0.999


### Magnitude of fractional part¶

q)x:6.13 -6.13
q)abs[x]
6.13 6.13
q)a-floor a:abs x
0.13 0.13


### Fractional part with sign¶

q)signum[x]*abs[x]mod 1
0.2 0.3 -0.2 -0.8 0 0 -0


### Leading digit of numeric code abbb¶

q)ld:{floor x%1000}
q)x:1319 8629 6581 6988 790 9045
q)ld x
1 8 6 6 0 9


### Last part of abbb¶

q)x:1234 5678 9012 345 6789
q)x mod 1000
234 678 12 345 789


## Infinity and beyond¶

### Identity for floating-point maximum, negative infinity¶

q)max 0#0.0
-0w
q)-1e100|-0w
-1e+100
q)-1e100|-0W
-2.147484e+09
q)-123456789|-0w
-1.234568e+08
q)-123456789|-0W
-123456789


### Identity for floating point minimum, positive infinity¶

q)min 0#0.0
0w
q)1e100&0w
1e+100
q)1e100&0W
2.147484e+09
q)123456789&0w
1.234568e+08
q)123456789&0W
123456789