1. ascending ordinals (ranking, shareable)

x:11 17 12 13 13 13 13 18

x[<x]?/:x

0 6 1 2 2 2 2 7

2. max scan x partition y

x:1 1 0 0 0 1 0 0 1 1

y:3 4 8 2 5 6 9 4 5 4

,/|\'(&x)_ y

3 4 8 8 8 6 9 9 5 4

3. min scan x partitions y

x:1 1 0 0 0 1 0 0 1 1

y:3 4 8 2 5 6 9 4 5 4

,/&\'(&x)_ y

3 4 4 2 2 6 6 4 5 4

4. are x and y permutations of each other

x:15 16 13 18 14 11 12

y:15 16 13 19 14 11 12

x[<x]~y[<y]

0

y:15 16 13 14 18 12 11

x[<x]~y[<y]

1

5. sort subvectors ascending

5a. sort ascending

sa:{x[<x]}

sa 30 10 40 20

10 20 30 40

sa"popfly"

"floppy"

5b. indices from lengths

il:{(#x)#+\0,x}

il 4 3

0 4

x:10 30 20 50 60 40 5

y:4 3

,/sa'(il y)_ x

10 20 30 50 5 40 60

6. subvector minima

x:1 1 0 0 0 1 1 0 0 1

y:3 4 8 2 5 6 9 4 5 4

&/'(&x)_ y

3 2 6 4 4

7. subvector grade up (see 15 for down)

x:1 0 0 1 0 0 1 0

y:14 12 18 16 13 15 11 17

{,/x+'<:'x _ y}[&x;y]

1 0 2 4 3 5 6 7

8. sort rows ascending

x: 3 5 _draw 10

x

(6 3 3 9 7

9 4 4 7 9

9 4 7 8 9)

sa'x

(3 3 6 7 9

4 4 7 9 9

4 7 8 9 9)

9. see 8

10. Replicate at depth; add new dimension y-fold after dimension x of array z

10a. depth (n is depth at which to apply f)

d:{[n;f]:[n;_f[n-1;f]';f]}

h:{d[x;y#,:]z}

z:2 5#"abcdefghij"

z

("abcde"

"fghij")

q:h[0;3;z]

^q

3 2 5

q

(("abcde"

"fghij")

("abcde"

"fghij")

("abcde"

"fghij"))

r:h[1;3;z]

^r

2 3 5

r

(("abcde"

"abcde"

"abcde")

("fghij"

"fghij"

"fghij"))

s:h[2;3;z]

^s

2 5 3

s

(("aaa"

"bbb"

"ccc"

"ddd"

"eee")

("fff"

"ggg"

"hhh"

"iii"

"jjj"))

11. merge x, y, z, … under control of g (mesh)

x:"abcd"

y:"123456789"

z:"zz"

g:15 _draw 3

g

1 0 1 1 2 1 2 1 1 0 1 0 1 0 0

(x,y,z)[<<g]

"3a45z6z78b9cd12"

12. see 11

13. ascending ordinals (ranking, all different)

x:15 16 13 18 15 12 13

<<x

3 5 1 6 4 0 2

14. subvector maxima

x:1 1 0 0 0 1 1 0 0 1

y:3 4 8 2 5 6 9 4 5 4

|/'(&x)_ y

3 8 6 9 4

15. subvector grade down (see 7 for up)

x:1 0 0 1 0 0 1 0

y:14 12 18 15 13 16 11 17

{,/x+'>:'x _ y}[&x;y]

2 0 1 5 3 4 7 6

16. merge x and y by g

x:5 9 8 7 4 3

y:10 20 30 40

g:1 0 1 1 0 0 1 0 1 1

(x,y)[<>g]

5 10 9 8 20 30 7 40 4 3

17. descending ordinals (ranking, all different)

x:15 16 13 18 14 11 12

<>x

2 1 4 0 3 6 5

18. sort ascending by internal alphabet

w:("once";"more";"into")

w

("once"

"more"

"into"")

w[<`\$w]

("into"

"more"

"once")

19. sort character matrix ascending

m:("scion";"icons";"coins")

m

("scion"

"icons"

"coins")

m[<m]

("coins"

"icons"

"scion")

20. is x a permutation

x:7 _draw -7

x

0 5 1 6 4 3 2

x~<<x

1

x:7 _draw 7

x

4 3 3 6 0 5 4

x~<<x

0

21. rotate infixes of y determined by boolean x to the left one place

y:"abcdefghij"

x:1 0 1 0 0 1 1 0 0 0

y[<x++\x]

22. index of first occurrence of minimum of x

x:5+13 _draw 10

x

14 12 10 9 6 12 6 11 8 12 6 13 6

*<x

4

x?&/x

4

23. index of first occurrence of maximum of x

x:5+13 _draw 10

x

11 10 8 8 8 7 5 9 12 12 10 12 8

x?|/x

8

*>x

8

24. median

x:11 _draw 100

x

61 20 51 12 34 31 51 29 35 17 89

x[<x]

12 17 20 29 31 34 35 51 51 61 89

x[(<x)[_.5*#x]]

34

25. Doubling quotes

f:{_ssr[x;"\"";"\"\""]}

a:"Did he say, \"Hello\"?"

f a

"Did he say, \"\"Hello\"\"?"

26. insert y "*" after "=" in x

x:"abc=,d=,fgh="

g:&x="="

g

3 6 11

y:5

(x,"*")[(#x)&<(!#x),(y*#g)#g]

"abc=*****,d=*****,fgh=*****"

27. insert 0 after indices y of x

x:"abc,def,gh"

y:(&x=","),#x

y

3 7 10

<(!#x),y

0 1 2 3 10 4 5 6 7 11 8 9 12

#x

10

(#x)><(!#x),y

1 1 1 1 0 1 1 1 1 0 1 1 0

28. insert g elements h after indices y of x

x:"abcd=,def=,gh="

y:&x="="

y

4 9 13

g:4

h:"x"

a:g*#y

a

12

a#y

4 9 13 4 9 13 4 9 13 4 9 13

(x,a#h)[<(!#x),a#y]

"abcd=xxxx,def=xxxx,gh=xxxx"

29. insert g elements h before indices y of x

x:"1234,234,34"

y:0 5 9

g:5

h:"*"

a:g*#y

((a#h),x)[<(a#y),!#x]

"*****1234,*****234,*****34"

30. grade up x according to key y

x:"fig lime"

y:" abcdefghijklmn"

y?/:x

6 9 7 0 12 9 13 5

<y?/:x

3 7 0 2 1 5 4 6

x[3 7 0 2 1 5 4 6]

" efgiilm"

31. merge

x:"egg"

y:"mrin"

g:1 0 1 0 1 1 0

(x,y)[<<g]

"merging"

32. sort ascending indices x according to data y

x:2 3 4 5 0 1 8 7 6

y:79 74 78 76 77 75 73 72 71

x[<y[x]]

8 7 6 1 5 3 4 2 0

33. sort matrix x on column y

x:5 6 _draw 100

x

(37 41 41 72 60 0

91 59 5 19 17 26

24 90 28 63 42 56

75 67 45 14 38 49

85 11 23 61 64 44)

y:2

x[;y]

41 5 28 45 23

<x[;y]

1 4 2 0 3

x[<x[;y];]

(91 59 5 19 17 26

85 11 23 61 64 44

24 90 28 63 42 56

37 41 41 72 60 0

75 67 45 14 38 49)

x:10 _draw 100

x

30 45 97 77 35 49 87 82 79 8

y:1

<x*1 -1[y]

2 6 7 8 3 5 1 4 0 9

y:0

<x*1 -1[y]

9 0 4 1 5 3 8 7 6 2

35. sort ascending

x:10 _draw 100

x

84 63 31 42 95 58 9 37 84 39

<x

6 2 7 9 3 5 1 0 8 4

x[<x]

9 31 37 39 42 58 63 84 84 95

36. sort y on x

x:6 _draw 10

y:6 _draw 20

x

9 2 3 1 9 3

y

7 8 17 11 16 6

y[<x]

11 8 17 6 7 16

37. invert permutation (the inverse puts y in ascending order)

x: 7 _draw -7

x

5 3 2 0 6 4 1

<x

3 6 2 1 5 0 4

check:

x[<x]

0 1 2 3 4 5 6

38. sort matrix descending

x:5 6 _draw 3

x

(0 0 0 1 0 1

1 0 2 2 0 1

0 2 1 1 1 1

0 0 0 1 1 2

1 2 2 0 2 1)

x[<x]

(0 0 0 1 0 1

0 0 0 1 1 2

0 2 1 1 1 1

1 0 2 2 0 1

1 2 2 0 2 1)

y:"abcde"[5 6 _draw 5]

y

("dcdbed"

"aaaaab"

"dcdbdc"

"baaace"

"eedbec")

y[<y]

("aaaaab"

"baaace"

"dcdbdc"

"dcdbed"

"eedbec")

39. reverse infixes in x of lengths y

x:11+!8

x

11 12 13 14 15 16 17 18

y:3 3 2

x[|>+\(!#x)_lin il y]

13 12 11 16 15 14 18 17

40. reverse infixes in x starting at indices y

x:1 0 1 0 0 1 0 0 0 1

y:1 2 3 4 5 6 7 8 9 10

+\x

1 1 2 2 2 3 3 3 3 4

>+\x

9 5 6 7 8 2 3 4 0 1

|>+\x

1 0 4 3 2 8 7 6 5 9

y[|>+\x]

2 1 5 4 3 9 8 7 6 10

41. indices of ones in boolean vector x

x:0 0 1 0 1 0 0 0 1 0

&x

2 4 8

42. move all blanks to end of text

x:" sign if i cant "

x[<" "=x]

"significant "

43. move elements of x with characteristic y to beginning

x:"mjinase"

y:0 1 0 0 1 1 0

x[>y]

"jasmine"

44. sort descending

x:8 _draw 10

x

3 5 0 4 5 2 4 5

x[>x]

5 5 5 4 4 3 2 0

45. binary representation of positive integer

x:16

2 _vs x

1 0 0 0 0

x:20

2 _vs x

1 0 1 0 0

46. transposed formatted integers 1 through x

x:15

q:10 _vs 1+!x

q

(0 0 0 0 0 0 0 0 0 1 1 1 1 1 1

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5)

"0123456789"[q]

("000000000111111"

"123456789012345")

47. polynomial with roots x

x:1 -6 8

{(x,0)-y*0,x}/1,x

1 -3 -46 48

x:2 4

{(x,0)-y*0,x}/1,x

1 -6 8

x:1 2

{(x,0)-y*0,x}/1,x

1 -3 2

x:1 2 3

{(x,0)-y*0,x}/1,x

1 -6 11 -6

x:(4 2 4 4 2 4

> 5 3 5 5 3 5

> 4 2 4 4 2 4

> 1 2 4 4 2 4

> 5 3 5 5 3 5

> 4 2 4 4 2 4)

48a. row minimum

rn:{x='&/'x}

rn x

(0 1 0 0 1 0

0 1 0 0 1 0

0 1 0 0 1 0

1 0 0 0 0 0

0 1 0 0 1 0

0 1 0 0 1 0)

48b. column maximum

cx:{x=\:|/x}

cx x

(0 0 0 0 0 0

1 1 1 1 1 1

0 0 0 0 0 0

0 0 0 0 0 0

1 1 1 1 1 1

0 0 0 0 0 0)

48c. minmax of rows and columns

minmax:{(rn x)&(cx x)}

minmax x

(0 0 0 0 0 0

0 1 0 0 1 0

0 0 0 0 0 0

0 0 0 0 0 0

0 1 0 0 1 0

0 0 0 0 0 0)

48d. locate 1s in ravel of Boolean matrix

ones:{&,/minmax x}

ones x

7 10 25 28

sp:{(^x) _vs ones x}

sp x

(1 1 4 4

1 4 1 4)

hex:"0123456789abcdef"

hd:{+hex[16 _vs x]}

x:10 12 19 1 28 100

hd x

("0a"

"0c"

"13"

"01"

"1c"

"64")

y:10 12 19 1 28 300

hd y

("00a"

"00c"

"013"

"001"

"01c"

"12c")

50. connectivity list from connectivity matrix

m:(1 0 1;1 0 1)

m

(1 0 1

1 0 1)

,/m

1 0 1 1 0 1

&,/m

0 2 3 5

(^m)_vs &,/m

(0 0 1 1

0 2 0 2)

lm:{(^x)_vs &,/x}

lm m

(0 0 1 1

0 2 0 2)

51. indices

ix:{(^x)_vs!*/^x}

ix[!6]

,0 1 2 3 4 5

ix[2 3#!6]

(0 0 0 1 1 1

0 1 2 0 1 2)

ix[2 3 2#!12]

(0 0 0 0 0 0 1 1 1 1 1 1

0 0 1 1 2 2 0 0 1 1 2 2

0 1 0 1 0 1 0 1 0 1 0 1)

52. truth table of order x

tt:{2 _vs !_2^x}

tt 1

,0 1

tt 2

(0 0 1 1

0 1 0 1)

tt 3

(0 0 0 0 1 1 1 1

0 0 1 1 0 0 1 1

0 1 0 1 0 1 0 1)

53. decimal digits from integer

dd:{10 _vs x}

dd 2001

2 0 0 1

dd 123456789

1 2 3 4 5 6 7 8 9

54. represent x in base y

x:256

y:16

y _vs x

1 0 0

x:36

y:2

y _vs x

1 0 0 1 0 0

x:123

y:10

y _vs x

1 2 3

55. indices in x containing elements in y

x:(3 7 8;2 5 9)

y:(3 1 6;8 9 2)

(^x)_vs &(,/x) _lin\: ,/y

(0 0 1 1

0 2 0 2)

hex:"0123456789abcdef"

x:" abcdef"

_ic x

32 97 98 99 100 101 102

16 _vs _ic x

(2 6 6 6 6 6 6

0 1 2 3 4 5 6)

hex[16 _vs _ic x]

("2666666"

"0123456")

+hex[16 _vs _ic x]

("20"

"61"

"62"

"63"

"64"

"65"

"66")

" ",'+hex[16 _vs _ic x]

(" 20"

" 61"

" 62"

" 63"

" 64"

" 65"

" 66")

,/" ",'+hex[16 _vs _ic x]

" 20 61 62 63 64 65 66"

x:"GOLDEN"

,/" ",'+hex[16 _vs _ic x]

"47 4f 4c 44 45 4e"

57. vector from date

_t

-1155413254

_gtime _t

19980522 35236

*_gtime _t

19980522

100000 100 100 _vs *_gtime _t

1998 5 22

58. pair each element of !x with each element of !y

x:3

y:4

(x,y)_vs !x*y

(0 0 0 0 1 1 1 1 2 2 2 2

0 1 2 3 0 1 2 3 0 1 2 3)

59. truth table of order x (see 52)

60. truth table of order x (see 52)

61. cyclic counter, repeating 1 through n

x:1+!10

y:8

1+x!y

2 3 4 5 6 7 8 1 2 3

62. integer and fractional parts of positive x

x:12.3 23.4 5.33 8.999

if:{(_ x),'x-_ x}

if x

((12;0.3)

(23;0.4)

(5;0.33)

(8;0.999))

63. represent x in radix 10 100 1000

x:123456

10 100 1000 _vs x

1 23 456

x:123456789

10 100 1000 _vs x

4 56 789

64. represent integer time hhmmss as character time, items separated by colons

64a. local time as integer of form hhmmss

x:_ltime _t

x

45628

q:2 _ \$ 1000 _sv 100 _vs 1000000 + x

q

"04056028"

@[q;2 5;:;":"]

"04:56:28"

65. represent integer date in form yyyymmdd as character date, parts separated by "/"

x:*_ltime _t

x

19980521

1000 _sv 10000 100 100 _vs x

1998005021

q:\$1000 _sv 10000 100 100 _vs x

q

"1998005021"

@[q;4 7;:;"/"]

"1998/05/21"

66. selection by encoded list

x:1 0 1

"abcdefgh"[2 _sv x]

"f"

x:0 0 0

"abcdefgh"[2 _sv x]

"a"

x:1 1 1

"abcdefgh"[2 _sv x]

"h"

67. Extrapolated value of abscissa x and ordinate y at g

x:-1 0 1

y:1 0 1.0 / y ~ x^2

g:10

|!#x

2 1 0

x^/:|!#x

(1 0 1.0

-1 0 1.0

1 1 1.0)

y _lsq x^/:|!#x

1 0 4.440892e-016

g _sv y _lsq x^/:|!#x

100.0

g:5

g _sv y _lsq x^/:|!#x

25.0

68. not relevant

69. value of ascending polynomial coefficients y at points x

x:-1 0 2

y: 4 0 5 1

x _sv\: y

-8 1 43

70. remove duplicate rows

x:("to"

"be"

"or"

"not"

"to"

"be")

?x

("to"

"be"

"or"

"not")

71. connectivity matrix from connectivity list

y

(0 1

0 2

1 0

1 2

2 2)

x

3

x _sv/:y

1 2 3 5 8

(!9)_lin x _sv/:y

0 1 1 1 0 1 0 0 1

(x,x)#(!x*x)_lin x _sv/:y

(0 1 1

1 0 1

0 0 1)

72. encode date as integer

x:*_ltime _t

x

19980522

73. remove trailing blanks

x:"trailing blanks "

73a. negative count of trailing blanks

nctb:{-+/&\|" "=x}

nctb x

-4

rtb:{(nctb x)_ x}

rtb x

"trailing blanks"

74. number of days in month x of Gregorian year y (ly from 463)

x:1

:[2=x

> 28+ly y

> (0,12#7#31 30)[x]]

31

x:2

:[2=x

> 28+ly y

> (0,12#7#31 30)[x]]

29

x:("ff";"a9";"8ac";"ffff")

x

("ff"

"a9"

"8ac"

"ffff")

"0123456789abcdef"?/:"ff"

15 15

"0123456789abcdef"?/:/:x

(15 15

10 9

8 10 12

15 15 15 15)

16 _sv/: "0123456789abcdef"?/:/:x

255 169 2220 65535

76. justify right

x:"trailing blanks "

76a. negative count of trailing blanks

nctb:{-+/&\|" "=x}

nctb x

-4

rj:{(nctb x)!x}

rj x

" trailing blanks"

x:("a ";"bc ";"d e ";"fg h ";"ij kl ";"mnopqr")

x

("a "

"bc "

"d e "

"fg h "

"ij kl "

"mnopqr")

rj'x

(" a"

" bc"

" d e"

" fg h"

" ij kl"

"mnopqr")

77. present value of cash flows c at times t and discount rate d

Example: a 3-year bond with an annual 10% coupon and discount rate of 0.9

pv:{[c;t;d]+/c*d^t}

c:0.1 0.1 1.1

t:1 2 3

d:0.9

pv[c;t;d]

0.9729

It is better to use current prices to derive a discount function

pv:{[c;t;d]+/c*d[t]}

pv[c;t;{[t] 0.9^t}]

0.9729

It is even more efficient to use the discount values directly

d: 0.9^0 1 2 3

d

1 0.9 0.81 0.729

pv[c;t;d]

0.9729

78. number from alphanumeric

x:"1998 51"

. x

1998 51

3 + . x

2001 54

79. index (1-origin) of first non-blank, counting from rear

x:"blanks at end "

(" "=x)

0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1

(" "=x) _sv 1

4

80. scattered indexing

x:2 3 4# _ci 97+!24

x

(("abcd"

"efgh"

"ijkl")

("mnop"

"qrst"

"uvwx"))

x ./:(0 0 0;1 1 3;1 2 2)

"atw"

81. raveled index from general index

x:2 3 4# _ci 97+!24

x[1;1;3]

"t"

^x

2 3 4

2 3 4 _sv 1 1 3

19

,//x

"abcdefghijklmnopqrstuvwx"

(,//x)[19]

"t"

82. future value of cash flows x at interest rate y

x:10 15 20 25

y:5

+/x*(1+y%100)^|!#x

74.11375

fv:{+/x*(1+y%100)^|!#x}

fv[x;y]

74.11375

83. see 69

84. scalar from boolean vector

x:1 0 0 1 1 1 0 1

2 _sv x

157

85. is matrix x antisymmetric

x:( 0 -7 1

7 0 -4

-1 4 0)

x~-+x

1

86. is matrix x symmetric

x:(0 4 7 1

4 8 6 4

7 6 2 0

1 4 0 6)

x~+x

1

x:4 4 _draw 10

x

(6 6 3 3

9 7 9 4

4 7 9 9

4 7 8 9)

x~+x

0

87. number of decimals (maximum 7)

nd:{:[1=#x;0;-2+#x]}

ff:{\$x-_ x}

ff 6.567

"0.567"

nd ff 1.234

3

nd ff 1234

0

nd ff 78.1234567

7

nd ff 78.12345678

7

"0.1234568"

88. name variable according to x

x:"test"

y:2 3#!6

. "var",(\$x),":y"

vartest

(0 1 2

3 4 5)

x:123

. "var",(\$x),":y"

var123

(0 1 2

3 4 5)

93. numbers from alphanumeric matrix

x:4 3#" 1 12 0.5"

x

(" 1"

" 12"

" "

"0.5")

x=" "

(1 1 0

1 0 0

1 1 1

0 0 0)

&/'x=" "

0 0 1 0

&&/'x=" "

,2

z:.:'x

z

(1;12;;0.5)

@[z;,2;:;0]

(1;12;0;0.5)

Putting it all together

@[.:'x;&&/'x=" ";:;0]

(1;12;0;0.5)

But there may be no all-blank rows

y=:4 3#" 1 123450.5"

y

(" 1"

" 12"

"345"

"0.5")

@[.:'y;&&/'y=" ";:;0]

(1;12;345;0.5)

na:{@[.:'x;&&/'x=" ";:;0]}

94. number from alphanumeric x, default y

x:""

y:"-1"

.((x~"")#"y"),x

"-1"

x:"234.5"

.((x~"")#"y"),x

234.5

95. numeric from proper alphanumeric nonnegative integer

x:"123 438"

d:"0123456789"

1 _ . "0 ",((#x)*&/x _lin d)#x

123 438

x:"123 45a 789"

1 _ . "0 ",((#x)*&/x _lin d)#x

0

96. conditional execution

@[+/;!6;:]

0 15

@[+/;"abc";:]

(1;"type")

98. execute rows of character matrix

x1:4

x2:9

x:2 5#"y1:x1y2:x2"

x

("y1:x1"

"y2:x2")

.:'x

(;)

y1

4

y2

9

99. numeric vector from evaluating rows of character matrix

x:2 5#"1+2 41+3 6"

x

("1+2 4"

"1+3 6")

.:'x

(3 5

4 7)

,/.:'x

3 5 4 7

100. indexing arbitrary rank array

x:2 3 4 5#!120

x[1]

((60 61 62 63 64

65 66 67 68 69

70 71 72 73 74

75 76 77 78 79)

(80 81 82 83 84

85 86 87 88 89

90 91 92 93 94

95 96 97 98 99)

(100 101 102 103 104

105 106 107 108 109

110 111 112 113 114

115 116 117 118 119))

x[0;0]

(0 1 2 3 4

5 6 7 8 9

10 11 12 13 14

15 16 17 18 19)

x[0;0;0]

0 1 2 3 4

x[0;0;0;0]

0

101. sum numbers in character matrix

x:\$!5

x

(,"0"

,"1"

,"2"

,"3"

,"4")

+/ .:' x

10

104. date ascending format

1 _ ,/".",/:\$|.:'0 4 6 _ \$* _ltime _t

"25.5.1998"

105. current time of 12-hour clock (AM & PM)

y:_t

y

-1155069892

x:*|_ltime y

x

201508

hms:{1 _,/":",/:1 _'\$100+100 _vs x!120000}

hms x

"08:15:08"

ap:{"AP"[115959<x],"M"}

ap x

"PM"

hm:{{(hms x)," ",ap x}[*|_ltime x]}

Ordinarily one would use _t as the argument to hm

hm x

"08:15:08 PM"

106. leading zeros for positive integers x in field width y

x:10 _draw 40

x

37 36 17 38 29 4 31 12 35 25

z:\$:'x+(y-1){x*10}/10

z

("1037"

"1036"

"1017"

"1038"

"1029"

"1004"

"1031"

"1012"

"1035"

"1025")

u:1 _/: z

u

("037"

"036"

"017"

"038"

"029"

"004"

"031"

"012"

"035"

"025")

107. current date, American

x:*_ltime _t

x

19980526

10000 100 100 _vs x

1998 5 26

1!10000 100 100 _vs x

5 26 1998

\$1!10000 100 100 _vs x

(,"5"

"26"

"1998")

" ",/:\$1!10000 100 100 _vs x

(" 5"

" 26"

" 1998")

,/" ",/:\$1!10000 100 100 _vs x

" 5 26 1998"

1 _ ,/" ",/:\$1!10000 100 100 _vs x

"5 26 1998"

"0123456789 "?/:1 _ ,/" ",/:\$1!10000 100 100 _vs x

5 10 2 6 10 1 9 9 8

d:"0123456789"

e:d,"/"

b:10000 100 100

k:{1 _ x}

s:" "

e[d?/:k ,/s,/:\$1!b _vs x]

"5/26/1998"

da:{e[d?/:k,/s,/:\$1!b _vs x]}

da[*_ltime _t]

"5/26/1998"

111. count of format of x

cf:{#\$x}

cf 12.345

6

cf -1

2

cf 1e-12

6

\$1e-12

"1e-012"

115, 116, 117. case structure

:[c0;t0;f]

:[c0;t0;c1;t1;f]

:[c0;t0;c1;t1;c2;t2;f]

:[c0;t0;c1;t1;c2;t2;c3;t3;f]

et cetera

In the first case, if c0 is nonzero, the result is t0; otherwise f.

In all cases, the result is the tn corresponding to the first nonzero cn.

If none of the cn are nonzero the result is f.

121. y-shaped array of numbers from x[0] to x[1]-1

x:4 9

y:3 4

(*x)+y _draw --/x

(5 8 7 4

8 7 5 8

8 7 7 5)

122. y objects selected with replacement from !x (roll)

y:3 5

x:7

y _draw x

(6 2 1 2 2

4 4 6 3 0

6 3 4 5 1)

123. y objects selected without replacement from !x (deal)

y:2 3

x:7

y _draw -x

(1 6 4

3 5 2)

6 _draw -6

3 0 1 5 4 2

123.1 normal deviates (from interval (0,1))

\p 4

x:4

x _draw 0

0.8683 0.5104 0.968 0.9831

124. predicted values of exponential fit

x:64 70 77 82 92

y:56 60 66 70 78

a:x^/:0 1

a

(1 1 1 1 1.0

64 70 77 82 92.0)

_log[y]

4 4.1 4.2 4.2 4.4

_lsq[_log[y];a]

3.3 0.012

_exp[_mul[+a;_lsq[_log[y];a]]]

56 60 66 70 78.0

125. predicted values of best linear fit (least squares)

x:64 70 77 82 92 107 125 143 165 189

y:56 60 66 70 78 88 102 118 136 155

a:x^/:0 1

a

(1 1 1 1 1 1 1 1 1 1.0

64 70 77 82 92 107 125 143 165 189.0)

_mul[+a;_lsq[y;a]]

55.32371 60.08021 65.62945 69.59319 77.52068 89.41191 103.6814 117.9509 135.3913 154.4173

This can also be written in infix form

(+a) _mul (y _lsq a)

55.32371 60.08021 65.62945 69.59319 77.52068 89.41191 103.6814 117.9509 135.3913 154.4173

126. g-degree polynomial fit of points (x,y)

x:64 70 77 82 92 107 125 143 165 189

y:(5*x^3)+(-1*x^2)+(4*x)+182

y

1307062 1710562 2277226 2750626 3885526 6114376 9750682 1.460134e+007 2.243424e+007 3.372156e+007

| _lsq[y;x^/:!g+1]

5 -1 4 182.0

|(y _lsq x^/:!g+1)

5 -1 4 182.0

127. coefficients of exponential fit of points (x,y)

x:64 70 77 82 92 107 125 143 165 189

y:56 60 66 70 78 88 102 118 136 155

a: _lsq[_log[y];x^/:0 1]

a

3.563398 0.00817742

a[0]: _exp[a[0]]

a

35.2829 0.00817742

a[0]*_exp[x*a[1]]

59.54624 62.54071 66.22511 68.98898 74.86758 84.63791 98.05964 113.6098 136.0024 165.4933

y

56 60 66 70 78 88 102 118 136 155.0

128. coefficients of best linear fit of points (x,y) (least squares)

x:64 70 77 82 92 107 125 143 165 189

y:56 60 66 70 78 88 102 118 136 155

z:_lsq[y;x^/:0 1]

z

4.587803 0.7927486

z[0]+z[1]*x / should be y, approximately

55.32371 60.08021 65.62945 69.59319 77.52068 89.41191 103.6814 117.9509 135.3913 154.4173

129. arctangent y%x

x:_sqrt[3]

y:1

_atan[y%x]

0.5235988

131. complementary angle (arccos sin x)

x:0.25

_acos[_sin x]

1.320796

x+_acos[_sin x] / should be 0.5*pi, approximately

1.570796

2*x+_acos[_sin x]

3.141593

132. rotation matrix for angle x (in radians) counter-clockwise

((_cos[x];-_sin[x]);(_sin[x];_cos[x]))

(0.9689124 -0.247404

0.247404 0.9689124)

x:0.5

57.295779513082323*x

28.64789

x:0.5

z:57.295779513082323*x

z

28.64789

0.017453292519943295*z

0.5

135. number of permutations of n objects taken k at a time

fac:{[n]:[n>1; n * _f[n-1];1]}

bin:{[n;k]fac[n]%fac[n-k]*fac[k]}

pn:{[n;k]fac[k]*bin[n;k]}

pn[5;3]

60.0

136. Pascal's triangle of order x (binomial coefficients) [see 1007]

137. Taylor series with coefficients y at point x

x:3

y:1 1 1

a:!#y

+/y*(x^a)%fac'[a]

8.5

y:30#1

x:1

a:!#y

+/y*(x^a)%fac'[a]

2.7182818308537429

139. Beta function

See gamma in appendix

Beta:{((gamma x)*gamma y)%gamma[x+y]}

142. number of combinations of n objects taken k at a time

fac:{[n]:[n>1; n * _f[n-1];1]}

bin:{[n;k]fac[n]%fac[n-k]*fac[k]}

bin[12;7]

792.0

bin[10;4]

210.0

143. indices of distinct items

x:"ajhajhja"

=x

(0 3 7

1 4 6

2 5)

144. histogram

x:13 _draw 12

x

8 3 11 9 9 4 6 6 3 3 9 7 9

h:{@[&1+|/x;x;+;1]}

b:h[x]

b

0 0 0 3 1 0 2 1 1 4 0 1

c:(1+|/b)-b

c

5 5 5 2 4 5 3 4 4 1 5 4

d:|+(b#\:1),'(c#\:0)

d

(0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 0

0 0 0 1 0 0 0 0 0 1 0 0

0 0 0 1 0 0 1 0 0 1 0 0

0 0 0 1 1 0 1 1 1 1 0 1)

" *"[d]

(" "

" * "

" * * "

" * * * "

" ** **** *")

145. count of x between endpoints (l,h)

x:5 5 _draw 100

l:10

h:80

x

(66 8 6 4 86

82 91 52 53 89

43 0 62 3 17

0 26 81 2 12

37 41 41 72 60)

xl:x>l

xl

(1 0 0 0 1

1 1 1 1 1

1 0 1 0 1

0 1 1 0 1

1 1 1 1 1)

xh:x<h

xh

(1 1 1 1 0

0 0 1 1 0

1 1 1 1 1

1 1 0 1 1

1 1 1 1 1)

xbetween:xl&xh

xbetween

(1 0 0 0 0

0 0 1 1 0

1 0 1 0 1

0 1 0 0 1

1 1 1 1 1)

+/'xbetween

1 2 3 2 5

146. compound interest for principals y at percentages g for periods x

x:1 2 3 4

y:1 2 3 4

g:0.5 1 1.5 2

z:y*\:(1+g%100)^\:x

\p 5

z

((1.005 1.01 1.0151 1.0202

1.01 1.0201 1.0303 1.0406

1.015 1.0302 1.0457 1.0614

1.02 1.0404 1.0612 1.0824)

(2.01 2.02 2.0302 2.0403

2.02 2.0402 2.0606 2.0812

2.03 2.0604 2.0914 2.1227

2.04 2.0808 2.1224 2.1649)

(3.015 3.0301 3.0452 3.0605

3.03 3.0603 3.0909 3.1218

3.045 3.0907 3.137 3.1841

3.06 3.1212 3.1836 3.2473)

(4.02 4.0401 4.0603 4.0806

4.04 4.0804 4.1212 4.1624

4.06 4.1209 4.1827 4.2455

4.08 4.1616 4.2448 4.3297))

147. locations of string x in string y (including overlaps)

x:"**"

y:"*abcugj**jy***kmhix**12"

ss:{z[&y[z+\:!#x]~\:x]}

ss[x;y;&((1-#x)_ y)=*x]

7 11 12 19

y[7 11 12 19+\:0 1]

("**"

"**"

"**"

"**")

148. node matrix from connection matrix (inverse to 157)

x:( 1 1 0 0 0

> 0 -1 0 1 1

> -1 0 1 -1 0

> 0 0 -1 0 -1)

Each column in matrix x represents a path between 2 nodes:

From node 0 to node 2

From node 0 to node 1

From node 2 to node 3

From node 1 to node 2

From node 1 to node 3

a:1 -1=\:x

a

((1 1 0 0 0

0 0 0 1 1

0 0 1 0 0

0 0 0 0 0)

(0 0 0 0 0

0 1 0 0 0

1 0 0 1 0

0 0 1 0 1))

b:_mul[a;!#x]

b

(0 0 2 1 1

2 1 3 2 3)

nc:{_mul[1 -1=\:x;!#x]}

nc x

(0 0 2 1 1

2 1 3 2 3)

149. number of decimals in x, maximum y

nd:{+/~0=(-_-x*10^y)!/:-_-(10^y)*10^-!y+1}

x:1.4321 1.21 10

y:3

nd[x;y]

3 2 0

10^-!y+1

1 0.1 0.01 0.001

-_-(10^y)*10^-!y+1

1000 100 10 1

-_-x*10^y

1433 1210 10000

(-_-x*10^y)!/:-_-(10^y)*10^-!y+1

(433 210 0

33 10 0

3 0 0

0 0 0)

~0=(-_-x*10^y)!/:-_-(10^y)*10^-!y+1

(1 1 0

1 1 0

1 0 0

0 0 0)

+/~0=(-_-x*10^y)!/:-_-(10^y)*10^-!y+1

3 2 0

150. sum items of x given by y

x:_log[1+!5]

x

0 0.6931472 1.098612 1.386294 1.609438

y:1 4 1 4 1

a:=y

a

(0 2 4

1 3)

+/'x[a]

2.70805 2.079442

+/x[0 2 4]

2.70805

+/x[1 3]

2.079442

+/'x[=y]

2.70805 2.079442

151. efficient execution of f x where x has repeated values

fx:{[f;x](f'x)(x?:)?/:x}

The phrase (x?:) is the same as (x:?x), which means x is assigned the unique elements of x. The phrase a?/:b means find each item of b in a. So (x?:)?/:x says find each item of x in the list of unique items of x (and while you're at it, reassign x to that list of unique items of x). The result is the list of indices of the original list x into the unique items of x.

x:1 2 3 2 3 2 1

?x

1 2 3

(?x)?/:x

0 1 2 1 2 1 0

(x?:)?/:x

0 1 2 1 2 1 0

This works because K's order of execution is right to left. the original value of x is "captured" before the (x?:) reassigns it.

x

1 2 3

f:{10*x}

f'x

10 20 30

The last bit is simply indexing:

(f'x)[0 1 2 1 2 1 0]

10 20 30 20 30 20 10

But K allows you to leave out the brackets:

(f'x)0 1 2 1 2 1 0

10 20 30 20 30 20 10

So the whole thing is

(f'x)(x?:)?/:x

152. sum items of y according by ordered codes g in x

y:1+!20

y

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

z:"abcde"

x:z[20 _draw #z]

x

"dcbbbaceeaeccbecbaea"

xz:z,x

yz:(&#z),y

xz

"abcdedcbbbaceeaeccbecbaea"

yz

0 0 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

+/'yz[=xz]

54 43 50 1 62

sc:{+/'((&#z),y)[=z,x]}

sc[x;y;z]

54 43 50 1 62

153. index of rows of y in corresponding rows of x

x:1+3 4#!12

x

(1 2 3 4

5 6 7 8

9 10 11 12)

y:(1 0 3 0

> 0 6 0 8

> 9 0 0 12)

x ?/:'y

(0 4 2 4

4 1 4 3

0 4 4 3)

154. range (nub; remove duplicate items)

x:"wirlsisl"

?x

"wirls"

?(1 2 3;4 5;1 2 3;4 5;1 2 3)

(1 2 3

4 5)

155. greatest common divisor of list x

gcd:{*|1+&&/'0=x!/:1+!&/x}

x:6 9 12

gcd x

3

&/x

6

!&/x

0 1 2 3 4 5

1+!&/x

1 2 3 4 5 6

x!/:1+!&/x

(0 0 0

0 1 0

0 0 0

2 1 0

1 4 2

0 3 0)

0=x!/:1+!&/x

(1 1 1

1 0 1

1 1 1

0 0 1

0 0 0

1 0 1)

&/'0=x!/:1+!&/x

1 0 1 0 0 0

&&/'0=x!/:1+!&/x

0 2

*|1+&&/'0=x!/:1+!&/x

3

156. classify y into x classes between min and max y

x:10

y:260 416 18 27 265 336 4

normalize y so minimum value is 0

sm:{x-&/x}

sm[y]

256 412 14 23 261 332 0

normalize again so values are in range 0 le y and y lt x

nr:{y*x%|/y}

nr[x;sm[y]]

6.21 10 0.34 0.558 6.33 8.06 0

compare against interior range boundaries, 1+!x-1

~nr[x;sm[y]]</:1+!x-1

(1 1 0 0 1 1 0

1 1 0 0 1 1 0

1 1 0 0 1 1 0

1 1 0 0 1 1 0

1 1 0 0 1 1 0

1 1 0 0 1 1 0

0 1 0 0 0 1 0

0 1 0 0 0 1 0

0 1 0 0 0 0 0)

count the number of boundaries passed by each

+/~nr[x;sm[y]]</:1+!x-1

6 9 0 0 6 8 0

157. connection matrix from node matrix (inverse to 148)

node matrix top and bottom rows give from and to nodes

x: (0 0 2 1 1

2 1 3 2 3)

enumerate count of range

!#?,/x

0 1 2 3

where is x equal to each of it

x=/:!#?,/x

((1 1 0 0 0

0 0 0 0 0)

(0 0 0 1 1

0 1 0 0 0)

(0 0 1 0 0

1 0 0 1 0)

(0 0 0 0 0

0 0 1 0 1))

subtract "to" matrix from "from" matrix

-/'x=/:!#?,/x

(1 1 0 0 0

0 -1 0 1 1

-1 0 1 -1 0

0 0 -1 0 -1)

158. see 20

159. is range of x 1 (are all items of x equal)

x:1 1 1 1 1

1=#?x

1

y:1 1 0 1 1

1=#?y

0

160. move blanks in x to end of list

x:"sign if i cant"

x[<x=" "]

"significant "

be:{x[<x=" "]}

y:("yo ho ho"

"and a bottle"

"of rum")

be'[y]

("yohoho "

"andabottle "

"ofrum ")

161. is y upper triangular

x:(1 0 0 1

> 0 2 1 0

> 0 0 1 2

> 0 0 0 0)

slt:{(!x)>\:!x}

slt[#x]

(0 0 0 0

1 0 0 0

1 1 0 0

1 1 1 0)

x*slt[#x]

(0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0)

zm:{(x,x)#0}

zm[#x]

(0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0)

iut:{zm[#x]~x*slt[#x]}

iut x

1

+x

(1 0 0 0

0 2 0 0

0 1 1 0

1 0 2 0)

iut[+x]

0

162. is x lower triangular

x:(1 0 0 0

0 2 0 0

0 1 1 0

1 0 2 0)

sut:{(!x)<\:!x}

sut[#x]

(0 1 1 1

0 0 1 1

0 0 0 1

0 0 0 0)

ilt:{zm[#x]~x*sut[#x]}

ilt[x]

1

ilt[+x]

0

163. polynomial product

x:1 2 1

y:1 3 3 1

y*/:x

(1 3 3 1

2 6 6 2

1 3 3 1)

1 _'zm[#x]

(0 0

0 0

0 0)

(1 _'zm[#x]),'y*/:x

(0 0 1 3 3 1

0 0 2 6 6 2

0 0 1 3 3 1)

(!#x)!'(1 _'zm[#x]),'y*/:x

(0 0 1 3 3 1

0 2 6 6 2 0

1 3 3 1 0 0)

+/(!#x)!'(1 _'zm[#x]),'y*/:x

1 5 10 10 5 1

pm:{+/(!#x)!'(1 _'zm[#x]),'y*/:x}

pm[x;y]

1 5 10 10 5 1

164. divisors

x:363

dv:{&0=x!/:!1+x}

dv[x]

1 3 11 33 121 363

dv 365

1 5 73 365

dv 367

1 367

dv'[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)

165. list of x zeros preceded by (y-x) ones

x:5

y:9

zo:{((y-x)#1),&x}

zo[x;y]

1 1 1 1 0 0 0 0 0

166. barchart of integer list x, down the page

x:2 5 7 4 9 3 6

xl:{(x#1),&y-x}

" X"[|+xl\:[x;|/x]]

(" X "

" X "

" X X "

" X X X"

" XX X X"

" XXXX X"

" XXXXXX"

"XXXXXXX"

"XXXXXXX")

bd:{" X"[|+xl\:[x;|/x]]}

bd[x]

(" X "

" X "

" X X "

" X X X"

" XX X X"

" XXXX X"

" XXXXXX"

"XXXXXXX"

"XXXXXXX")

167. list of x ones preceded by (y-x) xeros

x:3

y:9

xr:{(&y-x),x#1}

xr[x;y]

0 0 0 0 0 0 1 1 1

x:2 5 7 4 9 3 6

xr\:[x;y]

(0 0 0 0 0 0 0 1 1

0 0 0 0 1 1 1 1 1

0 0 1 1 1 1 1 1 1

0 0 0 0 0 1 1 1 1

1 1 1 1 1 1 1 1 1

0 0 0 0 0 0 1 1 1

0 0 0 1 1 1 1 1 1)

168. list of x zeros followed by (y-x) ones

x:3

y:9

(&x),(y-x)#1

0 0 0 1 1 1 1 1 1

zl:{(&x),(y-x)#1}

zl[x;y]

0 0 0 1 1 1 1 1 1

x:2 5 7 4 9 3 6

zl\:[x;y]

x:2 5 7 4 9 3 6

zl\:[x;y]

(0 0 1 1 1 1 1 1 1

0 0 0 0 0 1 1 1 1

0 0 0 0 0 0 0 1 1

0 0 0 0 1 1 1 1 1

0 0 0 0 0 0 0 0 0

0 0 0 1 1 1 1 1 1

0 0 0 0 0 0 1 1 1)

169. see 172

170. horizontal barchart of x with maximum z, normalized to length y

x:4

y:5

z:10

1

14

x:2 8 5 6 3 1 7 7 10 4

xl:{(x#1),&y-x}

(1 0 0 0 0

1 1 1 1 0

1 1 0 0 0

1 1 1 0 0

1 0 0 0 0

0 0 0 0 0

1 1 1 0 0

1 1 1 0 0

1 1 1 1 1

1 1 0 0 0)

("X "

"XXXX "

"XX "

"XXX "

"X "

" "

"XXX "

"XXX "

"XXXXX"

"XX ")

171. horizontal barchart of integer values x

(compare bh here with xl in 172)

bh:{@[&y;!x;:;1]}

" X"[bh\:[x;|/x]]

("XX "

"XXXXXXXX "

"XXXXX "

"XXXXXX "

"XXX "

"X "

"XXXXXXX "

"XXXXXXX "

"XXXXXXXXXX"

"XXXX ")

172. list of x ones followed by (y-x) zeros

x:5

y:9

(x#1),&(y-x)

1 1 1 1 1 0 0 0 0

xl:{(x#1),&y-x}

xl[x;y]

1 1 1 1 1 0 0 0 0

x:2 5 7 4 9 3 6

xl\:[x;y]

(1 1 0 0 0 0 0 0 0

1 1 1 1 1 0 0 0 0

1 1 1 1 1 1 1 0 0

1 1 1 1 0 0 0 0 0

1 1 1 1 1 1 1 1 1

1 1 1 0 0 0 0 0 0

1 1 1 1 1 1 0 0 0)

173. assign x to one of y classes of width h, starting with g

f:{[x;y;g;h] -1+ -1 _ #:'=(1+!y),-_-(x-g)%h}

x:32 56 36 48 36 24 28 44 52 32

y:4

h:10

g:10

f[x;y;g;h]

0 2 4 2

174. move x into first quadrant

sm:{x-&/x}

x:(1 6 4;3 4 7;7 8 6)

sm'[x]

(0 5 3

0 1 4

1 2 0)

175. primes to n

n:10

x:1+!n

x

1 2 3 4 5 6 7 8 9 10

x!/:x:1+!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)

+/0=x!/:x:1+!n

1 2 2 3 2 4 2 4 3 4

2=+/0=x!/:x:1+!n

0 1 1 0 1 0 1 0 0 0

&0,2=+/0=x!/:x:1+!n

2 3 5 7

pn:{[n]&0,2=+/0={x!/:x}1+!n}

pn 30

2 3 5 7 11 13 17 19 23 29

177. ordinal of word in x pointed at by y

ow:{+/~y<1+&x=" "}

x:"ordinal of word in x pointed at by y"

ow[x;5]

0

ow[x;6]

0

ow[x;7]

0

ow[x;8]

1

ow[x;26]

5

177. indices of start of string x in string y

x:"st"

y:"indices of start of string x in string y"

y _ss x

11 20 32

178. index of first occurrence of string x in string y

x:"st"

y:"index of first occurrence of string x in string y"

*y _ss x

12

179. contour levels y at points with altitude x

cl:{y[-1++/~y>x]}

y:-100 0 10 100 1000 10000

cl[-5;y]

-100

cl[0;y]

0

cl[99;y]

10

cl[9;y]

0

cl[10;y]

10

180. is x in range [y)

hc:{1 0~/:~x<\:y}

x:19 20 21 39 40 41

y:20 40

~x<\:y

(0 0

1 0

1 0

1 0

1 1

1 1)

hc[x;y]

0 1 1 1 0 0

181. which class of y x belongs to

cl:{-1++/x>/:y}

x:87 9 931 7 27 92 654 1416 7 911

y:0 50 100 1000

+/x>/:y

2 1 3 1 1 2 3 4 1 3

-1++/x>/:y

1 0 2 0 0 1 2 3 0 2

cl[x;y]

1 0 2 0 0 1 2 3 0 2

182. indices of consecutive repeated elements

x:"aaabccccdeee"

=x

(0 1 2

,3

4 5 6 7

,8

9 10 11)

#:'=x

3 1 4 1 3

&1<#:'=x

0 2 4

{x[&1<#:'x]}=x

(0 1 2

4 5 6 7

9 10 11)

re:{{x[&1<#:'x]}[=x]}

re x

(0 1 2

4 5 6 7

9 10 11)

183. maximum table

x:!10

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)

184. right justify fields x of length y to length g

x:"abcdefghij"

y:2 3 4 1

g:5

a:+\0,-1_ y

a

0 2 5 9

a _ x

("ab"

"cde"

"fghi"

,"j")

(g#" "),/:a _ x

(" ab"

" cde"

" fghi"

" j")

b:(-g)#/:(g#" "),/:a _ x

b

(" ab"

" cde"

" fghi"

" j")

,/b

rj:{[x;y;g],/(-g)#/:(g#" "),/:(+\0,-1 _ y) _ x}

rj[x;y;g]

" ab cde fghi j"

185. left justify fields x of length y to length g

x:"abcdefghij"

y:2 3 4 1

g:5

a:+\0,-1_ y

a

0 2 5 9

a _ x

("ab"

"cde"

"fghi"

,"j")

((:+\0,-1_ y) _ x),\:g#" "

("ab "

"cde "

"fghi "

"j ")

g#/:((:+\0,-1_ y) _ x),\:g#" "

("ab "

"cde "

"fghi "

"j ")

lj:{[x;y;g],/g#/L :+\0,-1_ y ),\:g#" " }

rj[x;y;g]

" ab cde fghi j"

lj:{[x;y;g],/g#/:((+\0,-1 _ y)_ x),\:g#" "}

lj[x;y;g]

"ab cde fghi j "

186. annuity coefficient for x periods at interest rate y%

x:10 15 20 25

y:8 9 10 15

+1-(1+y%100)^\:-x

(0.537 0.578 0.614 0.753

0.685 0.725 0.761 0.877

0.785 0.822 0.851 0.939

0.854 0.884 0.908 0.97)

ac:{(y%100)%/:+1-(1+y%100)^\:-x}

ac[x;y]

(0.149 0.156 0.163 0.199

0.117 0.124 0.131 0.171

0.102 0.11 0.117 0.16

0.0937 0.102 0.11 0.155)

187. direct matrix product

x:1+3 2#!6

y:1+2 4#!8

+:'x*\:\:y

+:'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)))

dp:{+:'x*\:\:y}

188. Shur product

x

(1 2

3 4

5 6)

y

(1 2 3 4

5 6 7 8)

((*|^x)#x)*(*^y)#'y

(1 4

15 24)

189. add x to each row of y

x:1+!4

y:3 4#2+!12

y

(2 3 4 5

6 7 8 9

10 11 12 13)

x+/:y

(3 5 7 9

7 9 11 13

11 13 15 17)

190. see 189

191. Shur sum

x

(1 2

3 4

5 6)

y

(1 2 3 4

5 6 7 8)

((*|^x)#x)+(*^y)#'y

(2 4

8 10)

192. add x to each column of y

x:1+!2

x

1 2

y:2 5#3+!10

y

(3 4 5 6 7

8 9 10 11 12)

x+'y

(4 5 6 7 8

10 11 12 13 14)

193. see 192

195. upper triangular matrix of order x

x:5

{~x>\:x}!x

(1 1 1 1 1

0 1 1 1 1

0 0 1 1 1

0 0 0 1 1

0 0 0 0 1)

ut:{{~x>\:x}[!x]}

ut 5

(1 1 1 1 1

0 1 1 1 1

0 0 1 1 1

0 0 0 1 1

0 0 0 0 1)

196. lower triangular matrix of order x

lt:{{~x<\:x}[!x]}

lt 5

(1 0 0 0 0

1 1 0 0 0

1 1 1 0 0

1 1 1 1 0

1 1 1 1 1)

197. identity matrix of order x

id:{(x,x)#1,x#0}

id 5

(1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1)

198. Hilbert matrix of order x

hm:{%1+(!x)+/:!x}

hm 5

(1 0.5 0.333 0.25 0.2

0.5 0.333 0.25 0.2 0.167

0.333 0.25 0.2 0.167 0.143

0.25 0.2 0.167 0.143 0.125

0.2 0.167 0.143 0.125 0.111)

199. multiplication table of order x

mt:{{x*\:x}[1+!x]}

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)

200. replicating a dimension of rank-3 array x y-fold

x:2 3 3#1+!18

y:3

x[;,/(y#1)*\:!(^x)[1];]

((1 2 3

4 5 6

7 8 9

1 2 3

4 5 6

7 8 9

1 2 3

4 5 6

7 8 9)

(10 11 12

13 14 15

16 17 18

10 11 12

13 14 15

16 17 18

10 11 12

13 14 15

16 17 18))

201. moving index y-wide for x

x:"abcdef"

y:3

(!(#x)-y-1)+\:y

3 4 5 6

202. indices of infixes of length y

x:4+!5

x

4 5 6 7 8

y:3

x+\:!y

(4 5 6

5 6 7

6 7 8

7 8 9

8 9 10)

203. one-column matrix from numeric list

x:7 _draw 100

x

34 31 51 29 35 17 89

+,x

(,34

,31

,51

,29

,35

,17

,89)

204. numeric array and its negative

x:3+3 4#!12

x

(3 4 5 6

7 8 9 10

11 12 13 14)

x,''-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))

205. remove trailing blank rows

x:+5 9#"abc de "

x

("aaaaa"

"bbbbb"

"ccccc"

" "

"ddddd"

"eeeee"

" "

" "

" ")

x~\:(#+x)#" "

0 0 0 1 0 0 1 1 1

~x~\:(#+x)#" "

1 1 1 0 1 1 0 0 0

||\|~x~\:(#+x)#" "

1 1 1 1 1 1 0 0 0

&||\|~x~\:(#+x)#" "

0 1 2 3 4 5

rtr:{x[&||\|~x~\:(#+x)#" "]}

rtr[x]

("aaaaa"

"bbbbb"

"ccccc"

" "

"ddddd"

"eeeee")

206. remove duplicate rows

x:("abc"

"def"

"abc"

"ghi"

"jkl"

"abc"

"ghi"

"abc")

?x

("abc"

"def"

"ghi"

"jkl")

207. indices in matrix x of rows of matrix y

x:+3 8#"abcdefgh"

x

("aaa"

"bbb"

"ccc"

"ddd"

"eee"

"fff"

"ggg"

"hhh")

y:+3 4#"afmc"

y

("aaa"

"fff"

"mmm"

"ccc")

x?/:y

0 5 8 2

208. see 206

209. remove trailing blank columns

x:3 9#"abc de "

x

("abc de "

"abc de "

"abc de ")

+rtr[+x]

("abc de"

"abc de"

"abc de")

x:3 9#" ed cba"

x

(" ed cba"

" ed cba"

" ed cba")

+|rtr[|+x]

("ed cba"

"ed cba"

"ed cba")

x:|+3 9#"abc de "

x

(" "

" "

" "

"eee"

"ddd"

" "

"ccc"

"bbb"

"aaa")

|rtr[|x]

("eee"

"ddd"

" "

"ccc"

"bbb"

"aaa")

213. maxima of infixes of x specified by boolean list y

x:-17 7 30 12 5 2 -5 6 -3 -19

y:10#1 1 0

y

1 1 0 1 1 0 1 1 0 1

|/x[&y]

12

214. see 159

215. see 159

216. rows of matrix x starting with y

x:("sit";"sat";"sin";"tin")

x

("sit"

"sat"

"sin"

"tin")

y:"si"

y _lin/:x

(1 1

1 0

1 1

0 1)

&/'y _lin/:x

1 0 1 0

&&/'y _lin/:x

0 2

rb:{x[&&/'y _lin/:x]}

rb[x;y]

("sit"

"sin")

217. index of last nonblank in string

x:("love's not "

"time's fool "

"though rosy ")

ln:{*|&~" "=x}

ln'x

9 10 10

218. single blank row from multiple

s:" a bc def g"

a:~s=" "

a

0 0 1 0 1 1 0 0 1 1 1 0 0 0 0 1

b:~a

b

1 1 0 1 0 0 1 1 0 0 0 1 1 1 1 0

c:>':0,b

c

1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0

d:a|c

d

1 0 1 1 1 1 1 0 1 1 1 1 0 0 0 1

e:&d

e

0 2 3 4 5 6 8 9 10 11 15

s[e]

" a bc def g"

x:+3 10#"a b c d"

x

("aaa"

" "

" "

"bbb"

" "

"ccc"

" "

" "

" "

"ddd")

a:x~\:(#+x)#" "

a

0 1 1 0 1 0 1 1 1 0

b:~a

c:>':0,a

c

0 1 0 0 1 0 1 0 0 0

d:b|c

d

1 1 0 1 1 1 1 0 0 1

e:&d

e

0 1 3 4 5 6 9

x[e]

("aaa"

" "

"bbb"

" "

"ccc"

" "

"ddd")

rs:{x[{&(~x)|xf[x]}x~\:(#+x)#" "]}

rs x

("aaa"

" "

"bbb"

" "

"ccc"

" "

"ddd")

219. see 147

220. remove duplicate blank columns

x:3 9#"a b c d"

x

("a b c d"

"a b c d"

"a b c d")

+rs[+x]

("a b c d"

"a b c d"

"a b c d")

221. is x an integer in interval [ g,h )

g:6

h:12

x:18

(x=_ x)&(~x<g)&(x<h)

0

x:7

(x=_ x)&(~x<g)&(x<h)

1

x:7.1

(x=_ x)&(~x<g)&(x<h)

0

222. maximum of x with weights y

x:1 2 3 4 5

y:5 4 3 2 1

|/x*y

9

223. minimum of x with weights y

x:1 2 3 4 5

y:5 4 3 2 1

&/x*y

5

224. extend distance table to next leg

x:( 0 50 80 20 999

> 50 0 20 40 30

> 80 20 0 999 30

> 20 40 999 0 10

> 999 30 30 10 0)

notice x[0;2] is 80 while x[0;1]+x[1;2] is 70

x(&/+)\:x

(0 50 70 20 30

50 0 20 40 30

70 20 0 40 30

20 40 40 0 10

30 30 30 10 0)

225. remove blank rows

x:("aaa"

> "bbb"

> " "

> "ccc"

> " ")

x _dv " "

("aaa"

"bbb"

"ccc")

226. remove blank columns

x:+4 4#"xxxx hhhh ii"

x

("x h "

"x h "

"x hi"

"x hi")

+(+x)_dv " "

("xh "

"xh "

"xhi"

"xhi")

227. see 69

228. is y a row of x

x:("xxx";"yyy";"zzz";"yyy")

x?"yyy"

1

229. see 228

230. extend a transitive binary relation

x:(0 1 0 0

> 0 0 1 1

> 1 0 0 0

> 0 0 1 0)

x(|/&)\:x

(0 0 1 1

1 0 1 0

0 1 0 0

1 0 0 0)

231. which rows of x contain elements different from y

x:("aaa";"bbb";"ooo";"pop")

y:"o"

~x=y

(1 1 1

1 1 1

0 0 0

1 0 1)

|/'~x=y

1 1 0 1

232. is y a row of x

x:("aaa"

"bbb"

"ooo"

"ppp"

"kkk")

y:"ooo"

y _in x

1

233. is x within range [ y )

x:9

y:(1 9

> 9 16

> 5 7

> 8 20

> 6 10)

x<'y

(0 0

0 1

0 0

0 1

0 1)

</'x<y

0 1 0 1 1

x(</<)/:y

0 1 0 1 1

234. is x within the range ( y ]

~x>'y

(0 1

1 1

0 0

0 1

0 1)

</'~x>'y

1 0 0 1 1

235. see 234

236. number of occurrences of x in y

y:3+7 _draw 6

y

6 4 7 7 6 6 4

x:7

+/x=y

2

x(+/=)/:y

0 0 1 1 0 0 0

237. average (mean) of x weighted by y

y:78 80 90 88 72

x:20 15 20 22 19

x*y

1560 1200 1800 1936 1368

+/x*y

7864

(+/x*y)%#x

1572.8

239. sum reciprocal series

x:10 9 10 7 8

y:80 63 70 63 64

+/y%x

39.0

240. matrix product

x:(1 2 3

> 4 5 6)

y:(1 2

> 3 4

> 5 6)

x _mul y

(22 28

49 64)

x(+/*)\:y

(22 28

49 64)

241. sum over subsets of x specified by y

x:1+3 4#!12

x

(1 2 3 4

5 6 7 8

9 10 11 12)

y:4 3#1 0

y

(1 0 1

0 1 0

1 0 1

0 1 0)

x _mul y

(4 6 4

12 14 12

20 22 20)

242. sum squares of x

x:1 2 3 4 5

+/x*x

55

243. dot product of vectors

x:1 2 3 4 5

y:10 20 30 40 50

+/x*y

550

244. product over subsets of x specified by y

x:1+3 4#!12

x

(1 2 3 4

5 6 7 8

9 10 11 12)

y:4 3#1 0

y

(1 0 1

0 1 0

1 0 1

0 1 0

x (*/^)\:y

(3 8 3.0

35 48 35.0

99 120 99.0)

245. randomize the random seed

_t

-1154371779

\r -1154371779

\r

-1154371779

246. see 242

247. interlace x[i]#1 and y[i]#0

x:1 3

y:2 4

a:,/+(x;y)

a

1 2 3 4

b:(#x,y)#1 0

b

1 0 1 0

c:a#'b

c

(,1

0 0

1 1 1

0 0 0 0)

d:,/c

d

1 0 0 1 1 1 0 0 0 0

,/(,/+(x;y))#'(#x,y)#1 0

1 0 0 1 1 1 0 0 0 0

248. center text x in line of width y

x:"1234567890"

y:16

a:y#x,y#" "

a

"1234567890 "

b:_ -0.5*y-#x

b

3

c:b!a

c

" 1234567890 "

ct:{(_ -0.5*y-#x)!y#x,y#" "}

ct[x;y]

" 1234567890 "

p:("1234567890";"1234";"123456";"1234567890123456")

p

("1234567890"

"1234"

"123456"

"1234567890123456")

ct\:[p;y]

(" 1234567890 "

" 1234 "

" 123456 "

"1234567890123456")

249. offset enumeration

x:10

y:3

x+!y

10 11 12

oi:{x+!y}

oi[x;y]

10 11 12

x:10 20 30

y:3 4 2

oi'[x;y]

(10 11 12

20 21 22 23

30 31)

,/oi'[x;y]

10 11 12 20 21 22 23 30 31

250. replicate y x times

x:3

y:10

x#y

10 10 10

rp:{x#y}

rp[x;y]

10 10 10

x:3 4 2

y:10 20 30

rp'[x;y]

(10 10 10

20 20 20 20

30 30)

,/rp'[x;y]

10 10 10 20 20 20 20 30 30

251. see 250

252. x alternate takes of 1s and 0s

x:1 2 3 4 5

b:(#x)#1 0

b

1 0 1 0 1

c:x#'b

c

(,1

0 0

1 1 1

0 0 0 0

1 1 1 1 1)

d:,/c

d

1 0 0 1 1 1 0 0 0 0 1 1 1 1 1

,/x#'(#x)#1 0

1 0 0 1 1 1 0 0 0 0 1 1 1 1 1

253. see 250

254. running parity of infixes of y indicated by x

x:1 0 0 0 0 1 0 0 0 0 1 0 0 0

y:1 0 0 1 1 1 0 0 1 0 1 1 0 0

a:&x

a

0 5 10

b:a _ y

b

(1 0 0 1 1

1 0 0 1 0

1 1 0 0)

c:(+\'b)!\:2

c

(1 1 1 0 1

1 1 1 0 0

1 0 0 0)

d:,/c

d

1 1 1 0 1 1 1 1 0 0 1 0 0 0

255. running sum of infixes of y indicated by x

x:1 0 0 0 1 0 0 0 1

y:1 2 3 4 5 6 7 8 9

a:&x

a

0 4 8

b:a _ y

b

(1 2 3 4

5 6 7 8

,9)

c:+\'b

c

(1 3 6 10

5 11 18 26

,9)

d:,/c

d

1 3 6 10 5 11 18 26 9

rs:{,/+\'(&x)_ y}

rs[x;y]

1 3 6 10 5 11 18 26 9

256. groups of 1s in y pointed at by x

y

0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1

x

0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1

a:+\>':0,y

a

0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 2

y&a=|\x*a

0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1

257. sums of infixes of x determined by lengths y

x:1+!10

x

1 2 3 4 5 6 7 8 9 10

y:2 3 2 3

a:+\0,-1 _ y

a

0 2 5 7

a _ x

(1 2

3 4 5

6 7

8 9 10)

+/'a _ x

3 12 13 27

259. removing leading and trailing blanks

x:" abcd e fg "

a:~x=" "

a

0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0

(|\a)&(||\|a)

0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0

&(|\a)&(||\|a)

3 4 5 6 7 8 9 10 11 12

x[&(|\a)&(||\|a)]

"abcd e fg"

lt:{x[&(|\a)&||\|a:~x=" "]}

lt x

"abcd e fg"

260. first 10 figurate numbers of order x

fg:{x+\/10#1}

fg 0

1 1 1 1 1 1 1 1 1 1

fg 1

1 2 3 4 5 6 7 8 9 10

fg 2

1 3 6 10 15 21 28 36 45 55

fg 3

1 4 10 20 35 56 84 120 165 220

fg 4

1 5 15 35 70 126 210 330 495 715

261. first group of 1s

x:1 1 1 0 1 0 1

x&&\x=|\x

1 1 1 0 0 0 0

x:0 0 0 1 1 0 1

x&&\x=|\x

0 0 0 1 1 0 0

x:0 1 0 1 0 1 0

x&&\x=|\x

0 1 0 0 0 0 0

262. value of saddle point (see 48)

x:(5 4 6 4 12 5

> 16 2 4 5 16 18

> 8 18 7 12 16 11

> 20 17 16 14 16 20

> 16 8 12 9 17 13)

rn:{x='&/'x}

cx:{x=\:|/x}

minmax:{(rn x)&(cx x)}

minmax x

(0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 1 0 0

0 0 0 0 0 0)

ones:{&,/minmax x}

ones x

,21

(,/x)[ones[x]]

,14

264. insert x[i] blanks after y[g[i]]

b:(0,g)_ y

b

("ab"

"cd"

"ef"

,"g")

c:b,'(x,0)#\:" "

c

("ab "

"cd "

"ef "

,"g")

,/c

"ab cd ef g"

ib:{[x;y;g],/((0,g)_ y),'(x,0)#\:" "}

ib[x;y;g]

"ab cd ef g"

265. insert x[i] zeroes after i-th infix of y

y:0 0 1 0 1 0 1 1

x:1 2 2 1

&y

2 4 6 7

a:@[&#y;&y;:;x]

a

0 0 1 0 2 0 2 1

b:1+a

b

1 1 2 1 3 1 3 2

d:(!(#y)++/x)

d

0 1 2 3 4 5 6 7 8 9 10 11 12 13

d:(1_!(#y)++/x)

d

1 2 3 4 5 6 7 8 9 10 11 12 13 14

d _lin c

1 1 0 1 1 0 0 1 1 0 0 1 0 1

266. remove trailing blanks

x:" phrase 266 "

a:~x=" "

a

0 0 1 1 1 1 1 1 0 1 1 1 0 0 0

b:||\|a

b

1 1 1 1 1 1 1 1 1 1 1 1 0 0 0

x[&b]

" phrase 266"

x:" phrase 267 "

a:~x=" "

a

0 0 1 1 1 1 1 1 0 1 1 1 0 0

b:|\a

b

0 0 1 1 1 1 1 1 1 1 1 1 1 1

x[&b]

"phrase 267 "

268. is x in ascending order

x:2 5 7 9 6 8 3

x~x[<x]

0

x:0 1 1 1 7 8 9

x~x[<x]

1

269. see 248

270. see 268

271. slight variation of 264

272. see 266

273. join scalar to each list item

x:"a"

y:"01234"

x,/:y

("a0"

"a1"

"a2"

"a3"

"a4")

x:("01234";"56789")

y:("abcde";"fghij")

x,y

("01234"

"56789"

"abcde"

"fghij")

x,'y

("01234abcde"

"56789fghij")

274. see 273

275. see 76

276. see 185

277. end indicators from lengths

x:1 2 3 4 5

+\x

1 3 6 10 15

-1++\x

0 2 5 9 14

+/x

15

@[&+/x;-1++\x;:;1]

1 0 1 0 0 1 0 0 0 1 0 0 0 0 1

278. start indicators from lengths

x:1 2 3 4 5

(!+/x) _lin\: +\0,x

1 1 0 1 0 0 1 0 0 0 1 0 0 0 0

279. variation of 265

280. see 41

281. value of Taylor series with coefficients y at x

x:12

y:7 5 6 6

1+!-1+#y

1 2 3

1.0,x%1+!-1+#y

1 12 6 4.0

*\1.0,x%1+!-1+#y

1 12 72 288.0

y**\1.0,x%1+!-1+#y

7 60 432 1728.0

+/y**\1.0,x%1+!-1+#y

2227.0

282. index of first blank

x:"ab c d"

x?" "

2

x:("ab c d";" a bc";"abcd ")

x

("ab c d"

" a bc"

"abcd ")

x?\:" "

2 0 4

283. locate field y of fields beginning with first of x

y:2

y=+\x=*x

0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0

x[&y=+\x=*x]

"abbb"

y:4

x[&y=+\x=*x]

284. sum items of x marked by y

x:1 2 3 4 5 6 7

y:1 1 1 2 2 3 3

=y

(0 1 2

3 4

5 6)

x[=y]

(1 2 3

4 5

6 7)

+/'x[=y]

6 9 13

y:1 2 1 3 3 2 1

+/'x[=y]

11 8 9

285. moving sum

y:3

x:1 2 3 4 5

+\x

1 3 6 10 15

(-y)

-3

(-y)_ a:+\x

1 3

0,(-y)_ a

0 1 3

(y-1)_ a

6 10 15

((y-1)_ a)-0,(-y)_ a

6 9 12

ms:{((y-1)_ a)-0,(-y)_ a:+\x}

ms[x;y]

6 9 12

286. FIFO stock y decremented with x units

x:5

y:1 2 3 4 5

(+\y)-x

-4 -2 1 5 10

g:0|(+\y)-x

g

0 0 1 5 10

-':0,g

0 0 1 4 5

ff:{-':0,0|(+\y)-x}

ff[x;y]

0 0 1 4 5

289. or-scan of infixes of y indicated by x

y:1 0 0 1 0 1 0 0

x:1 0 1 0 0 0 1 0

a:&x

b:a _ y

b

(1 0

0 1 0 1

0 0)

c:|\'b

c

(1 1

0 1 1 1

0 0)

,/c

1 1 0 1 1 1 0 0

290. and-scan of infixes of y indicated by x

y:1 0 0 1 0 1 0 0

x:1 0 1 0 0 0 1 0

a:&x

b:a _ y

b

(1 0

0 1 0 1

0 0)

c:&\'b

c

(1 0

0 0 0 0

0 0)

,/c

1 0 0 0 0 0 0 0

291. sums of infixes of y indicated by x

y:1 2 3 4 5

x:1 0 1 0 1

a:&x

b:a _ y

b

(1 2

3 4

,5)

c:+/'b

c

3 7 5

292. groups of 1s in y pointed to by x

y:1 1 1 0 0 1 1

x:0 1 0 1 0 0 0

-1 _ 0,y

0 1 1 1 0 0 1

+\y>-1 _ 0,y

1 1 1 1 1 2 2

x&y

0 1 0 0 0 0 0

a:+\y>-1 _ 0,y

a

1 1 1 1 1 2 2

a[&x&y]

,1

a _lin ,1

1 1 1 1 1 0 0

y&a _lin ,1

1 1 1 0 0 0 0

go:{y&a _lin(a:+\y>-1 _ 0,y)[&x&y]}

go[x;y]

1 1 1 0 0 0 0

293. locate quotes and text between them

x:"abc\"de\"f"

#x

8

a:x="\""

a

0 0 0 1 0 0 1 0

b:&a

b

3 6

c:(*b)+!1+--/b

c

3 4 5 6

@[&#x;c;:;1]

0 0 0 1 1 1 1 0

294. locate text between quotes

x:"abc\"de\"f"

#x

8

a:x="\""

a

0 0 0 1 0 0 1 0

b:&a

b

3 6

c:b+1 -1

c

4 5

d:(*c)+!1+--/c

d

4 5

@[&#x;d;:;1]

0 0 0 0 1 1 0 0 c

295. depth of parentheses

x:"a(b((c)de)f)g(h)"

dp:{+\("("=x)--1 _ 0,")"=x}

dp x

0 1 1 2 3 3 3 2 2 2 1 1 0 1 1 1

x:"a(b((cde)f)g(ki)"

dp x

0 1 1 2 3 3 3 3 3 2 2 1 2 2 2 2

x:"ab((c)de)f)g(ki)"

dp x

0 0 1 2 2 2 1 1 1 0 0 -1 0 0 0 0

x:"a(b((c)de)f)g(h)"

dp x

0 1 1 2 3 3 3 2 2 2 1 1 0 1 1 1

296. starting positions of infixes from lengths x

x:2 3 1 5

+\-1 _ 0,x

0 2 5 6

sl:{+\-1 _ 0,x}

sl[x]

0 2 5 6

x:"abcdef"

y:1 1 0 0 1 0

a:&y

a

0 1 4

b:#:'a _ x

b

1 3 2

c:b#'a

c

(,0

1 1 1

4 4)

d:,/c

d

0 1 1 1 4 4

x[d]

"abbbee"

x[,/(#:'a _ x)#'a:&y]

"abbbee"

sh:{x[,/(#:'a _ x)#'a:&y]}

sh[x;y]

"abbbee"

298. see 266

299. see 267

300. gth infix of y marked by x

x:1 0 0 1 0 1 0 0 0 1 0

y:"abcdefghijk"

g:2

a:&x

a

0 3 5 9

b:a _ x

b

("abc"

"de"

"fghi"

"jk")

b[g]

"fghi"

((&x)_ y)[g]

"fghi"

301. alternating sum series

x:1+!10

x

1 2 3 4 5 6 7 8 9 10

a:((#x)#1 -1)

a

1 -1 1 -1 1 -1 1 -1 1 -1

b:x*a

+\b

1 -1 2 -2 3 -3 4 -4 5 -5

as:{+\x*(#x)#1 -1}

as[x]

1 -1 2 -2 3 -3 4 -4 5 -5

302. x first triangular numbers

x:6

+\!x

0 1 3 6 10 15

303. smearing 1s between pairs of 1s

x:0 1 0 0 1 0 1 0 1 0 1 1 0

a:(+\x)!2

a

1 1 1 0 0 0 0 1 1 0 0

x|a

1 1 1 1 0 0 0 1 1 1 0

x|(+\x)!2

1 1 1 1 0 0 0 1 1 1 0

304. invert 0s following 1st 1

x:0 0 1 0 0 1 1

|\x

0 0 1 1 1 1 1

305. invert fields marked by pairs of 1s

x:1 0 1 0 0 1 0 0 1

a:(+\x)!2

a

1 1 0 0 0 1 1 1 0

~x

0 1 0 1 1 0 1 1 0

(~x)&a

0 1 0 0 0 0 1 1 0

(~x)&(+\x)!2

0 1 0 0 0 0 1 1 0

306. invert all 1s after 1st 0

x:1 1 0 1 0 1 0

&\x

1 1 0 0 0 0 0

307. invert all 1s after 1st 1

x:0 0 1 1 0 1

&#x

0 0 0 0 0 0

x?1

2

@[&#x;x?1;:;1]

0 0 1 0 0 0

308. invert all 0s after 1st 0

x:1 0 0 1 1 0

a:x?0

a

1

!#x

0 1 2 3 4 5

b:(a+1)_!#x

b

2 3 4 5

@[x;b;:;1]

1 0 1 1 1 1

@[x;(1+x?0)_!#x;:;1]

1 0 1 1 1 1

309. running parity

x:0 1 1 1 1 0 1 0 0

(+\x)!2

0 1 0 1 0 0 1 1 1

310. running sum

x:1 20 300 4000

+\x

1 21 321 4321

311. see 159

312. maximum separation of items of x

x:10+7 _draw 9

x

17 14 14 17 14 17 18

(|/x)-(&/x)

4

313. value of two-by-two determinant

det: {-/y*|x}

x:(13 21;34 55)

x

(13 21

34 55)

(13 * 55) - (34 * 21)

1

13 21 * 55 34

715 714

-/13 21 * 55 34

1

det x

,1

314. see 313

315. see 159

316. see 159

317. see 159

318. area of triangle with sides x (Heron's rule)

x:3 4 5

hr:{(*/(+/x%2)-0,x)^0.5}

hr[x]

6.0

319. standard deviation

x:44 77 48 24 28 36 17 49 90 91

std:{((+/(x-(+/x)%#x)^2)%#x)^0.5}

std[x]

25.48411

320. variance (dispersion)

x:44 77 48 24 28 36 17 49 90 91.0

var:{(+/(x-(+/x)%#x)^2)%#x}

var[x]

649.44

(var[x])^0.5

25.48411

321. y-th moment of x

x:44 77 48 24 28 36 17 49

ym:{(+/(x-(+/x)%#x)^y)%#x}

ym[x;2]

309.23

ym[x;0]

1.0

ym[x;1]

4.4409e-016

ym[x;3]

3889.9

322.

323. see 248

324. see 159

325. average (mean)

av:{(+/x)%#x}

av[1 10 100]

37.0

326. see 325

327. see 70

328. number of items

#"abcd"

4

#(1;2 3;4 5 6)

3

#2 3 4#!24

2

329. mask from positive integers in x

x:-5+7 _draw 10

x

2 3 3 -2 4 4 -1

(!1+|/x) _lin x

0 0 1 1 1

330. index of 1st occurrence of maximum item of x

x:5 3 7 0 5 7 2

x?|/x

2

331. identity for floating point maximum, negative infinity -0i

-1e100|-0i

-1e+100

identity for integer maximum, negative infinity -0I

-123456789|-0I

-123456789

332. see 326

333. quick membership for nonnegative integers

x:5 3 7 2

y:8 5 2 6 1 9

&1+|/x,y

0 0 0 0 0 0 0 0 0 0

a:&1+|/x,y

@[a;y;:;1]

0 1 1 0 0 1 1 0 1 1

(@[a;y;:;1])[x]

1 0 0 1

@[&1+|/x,y;y;:;1][x]

1 0 0 1

334. nonnegative maximum

x:1 2 3 4 5

|/x,0

5

x:-1 -2 -3 -4 -5

|/x,0

0

x:!0

x

!0

|/x,0

0

335. maximum

x:5 3 7 2

|/x

7

336. index of first occurrence of minimum

x:5 3 7 2 5 3

x?&/x

3

337. identity for floating point minimum, positive infinity 0i

1e100&0i

1e+100

identity for integer minimum, positive infinity 0I

123456789&0I

123456789

338. locate first occurrence in x of an item of y

x:"abcdef"

y:"dbf"

&/x?/:y

1

x[1]

"b"

339. minimum

x:5 3 7 2

&/x

2

340. see 159

341. see 159

342. arabic from roman number

x:"MCMIX"

"MDCLXVI"?/:x

0 2 0 6 4

a:0,1000 500 100 50 10 5 1["MDCLXVI"?/:x]

a

0 1000 100 1000 1 10

a<1!a

1 0 1 0 1 0

_ a*-1^a<1!a

0 1000 -100 1000 -1 10

+/_ a*-1^a<1!a

1909

ar:{+/_ a*-1^a<1!a:0,1000 500 100 50 10 5 1["MDCLXVI"?/:x]}

ar[x]

1909

343. see 159

344. pairwise match

x:("123";"123";"45";"45")

x

("123"

"123"

"45"

"45")

~':x

1 0 1

(~':x),0

1 0 1 0

pm:{(~':x),0}

pm 1 1 1 1 2 2 3 4 4 4

1 1 1 0 1 0 0 1 1 0

345. do ranges of x and y match

x:1 2 3

y:3 2 1 1

a:?x

x

1 2 3

a

1 2 3

b:?y

b

3 2 1

a[<a]~b[<b]

1

om:{x[<x]~y[<y]}

om[?x;?y]

1

om[?"bca";?"cabba"]

1

346. see 20

347. see 159

348. do x and y have items in common

x:"aba"

y:"cdeac"

x _lin\: y

1 0 1

|/x _lin\: y

1

y:"edge"

|/x _lin\: y

0

349. see 159'

350. is x 1s and 0s only (Boolean)

x:0 1 0 1

&/x _lin\:0 1

1

x:1 1 1 1

&/x _lin\:0 1

1

x:1 0 1 2

&/x _lin\:0 1

0

351. is x a subset of y

x:"abgk"

y:"abcdefghijkl"

&/x _lin\:y

1

x:"abgx"

&/x _lin\:y

0

352. see 159

353. are items unique

x:"abcdefg"

(#x)=(#?x)

1

x:"abcdefa"

(#x)=(#?x)

0

354. see 159

355. none

x:&7

x

0 0 0 0 0 0 0

~|/x

1

x:7#0 1

x

0 1 0 1 0 1 0

~|/x

0

356. any

x:&7

|/x

0

x:7#0 1

|/x

1

357. does x match y

x:("abc";`sy;1 3 -7)

y:("abc";`sy;1 3 -7)

x~y

1

x:1 2 3

y:1 4 3

x~y

0

358. see 159

359. locate blank rows

x:+5 6#"a bc d"

x

("aaaaa"

" "

"bbbbb"

"ccccc"

" "

"ddddd")

x~\:(#+x)#" "

0 1 0 0 1 0

360. all

x:1 1 0 1

&/x

0

x:1 1 1 1

&/x

1

x:0 0 0 0

&/x

0

361. parity

x:2 _vs !8

x

(0 0 0 0 1 1 1 1

0 0 1 1 0 0 1 1

0 1 0 1 0 1 0 1)

(+/x)!2

0 1 1 0 1 0 0 1

362. count of occurrences of x in y

x:"q"

y:"quaquaqua"

+/x=y

3

qu:{(q%x),(z%q:qq[x;y;z])}

qq:{-0.5*y+sg[y]*ds[x;y;z]}

ds:{_sqrt[(y*y)-(4*x*z)]}

sg:{(x>0)-(x<0)}

a:1

b:-1e30

c:1

sg[b]

-1

ds[a;b;c]

1e+030

qq[a;b;c]

1e+030

qu[a;b;c]

1e+030 1e-030

qu[1;-8;15]

5 3.0

364. see 325

365. see 325

366. count of scalars

cs:{#,//x}

cs[1]

1

cs[1 2]

2

cs[(1 2;3 4 5)]

5

cs[(1 2;(3 4;5))]

5

cs[("ab";("cd";"efg"))]

7

cs[!0]

0

367. alternating product

x:1 2 3 4 5

a:(#x)#1 -1

a

1 -1 1 -1 1

*/x^a

1.875

368. product

x:1 2 3 4 5

*/x

120

369. alternating sum

x:1+!10

x

1 2 3 4 5 6 7 8 9 10

a:((#x)#1 -1)

a

1 -1 1 -1 1 -1 1 -1 1 -1

b:x*a

+/b

-5

+/x*(#x)#1 -1

-5

370. count of 1s in Boolean list

x:1 0 0 1 0 1 1

+/x

4

371. scalar from 1-item list

x:,5

x

,5

+/x

5

a:,"a"

a

,"a"

+/a

type error

+/a

^

> \

a[0]

"a"

372. sum columns of matrix

x:1+3 4#!12

x

(1 2 3 4

5 6 7 8

9 10 11 12)

+/x

15 18 21 24

373. sum rows of matrix

x:1+3 4#!12

x

(1 2 3 4

5 6 7 8

9 10 11 12)

+/'x

10 26 42

374. sum

x:1 2 3 4 5

+/x

15

375. insert row x in matrix y after row g

y:4 3#1+!12

y

(1 2 3

4 5 6

7 8 9

10 11 12)

x:13 14 15

g:2

a:y,,x

a

(1 2 3

4 5 6

7 8 9

10 11 12

13 14 15)

b:<(!#y),g

b

0 1 2 4 3

a[b]

(1 2 3

4 5 6

7 8 9

13 14 15

10 11 12)

(y,,x)[<(!#y),g]

(1 2 3

4 5 6

7 8 9

13 14 15

10 11 12)

376. append y at the bottom of matrix x

x:4 3#1+!12

x

(1 2 3

4 5 6

7 8 9

10 11 12)

y:13 14 15

x,,y

(1 2 3

4 5 6

7 8 9

10 11 12

13 14 15)

377. fill x to length y with x's last item

x:"quiz"

y:9

a:(!y)&-1+#x

a

0 1 2 3 3 3 3 3 3

x[a]

"quizzzzzz"

x[(!y)&-1+#x]

"quizzzzzz"

or, an alternative way:

y#x,y#*|x

"quizzzzzz"

378. [omitted]

379. remove leading, multiple and trailing y's from x

x:0 0 1 2 0 0 3 4 0 5 0 0 0

y:0

a:x=y

b:~a&1!a

a

1 1 0 0 1 1 0 0 1 0 1 1 1

b

0 1 1 1 0 1 1 1 1 1 0 0 0

x[&b]

0 1 2 0 3 4 0 5

a[0]_ x[&b]

1 2 0 3 4 0 5

380. change items of x with value y[0] to y[1]

x:"abcde"[15 _draw 5]

x

y:"d "

@[x;&x=*y;:;*|y]

" aeeca bbcbe c"

381. invert all but 1st 1 in group of 1s

x:0 0 1 1 1 0 1 1 0 1

x>-1 _ 0,x

0 0 1 0 0 0 1 0 0 1

m381:{x>-1 _ 0,x}

m381[x]

0 0 1 0 0 0 1 0 0 1

382. insert y in x after index g

x:1 2 3

y:10*1+!7

y

10 20 30 40 50 60 70

g:3

((g+1)#y),x,(g+1)_ y

10 20 30 40 1 2 3 50 60 70

383. pairwise difference

x:9 3 5 2 0

--':x

6 -2 3 2

384. drop 1st, postpend 0

x:3 4 5 6

1 _ x,0

4 5 6 0

385. drop last, prepend 0

x:3 4 5 6

-1 _ 0,x

0 3 4 5

386. shift x right y, fill 0

x:1+!12

x

1 2 3 4 5 6 7 8 9 10 11 12

y:3

@[(-y)!x;!y;:;0]

0 0 0 1 2 3 4 5 6 7 8 9

387. shift x left y, fill 0

x:1+!12

x

1 2 3 4 5 6 7 8 9 10 11 12

y:3

@[y!x;((#x)-y)+!y;:;0]

4 5 6 7 8 9 10 11 12 0 0 0

388. drop y rows from top of matrix x

x:6 3#!18

x

(0 1 2

3 4 5

6 7 8

9 10 11

12 13 14

15 16 17)

y:2

y _ x

(6 7 8

9 10 11

12 13 14

15 16 17)

389. Playing order of x ranked players

i:{1+2_sv'+|tt[-_-log2[x]]}

j:{@[x;&x>y;:;0]}

k:{j[i[x];x]}

x:6

i[x]

1 5 3 7 2 6 4 8

j[i[x];x]

1 5 3 0 2 6 4 0

k[x]

1 5 3 0 2 6 4 0

390. Conform table x rows to list y

f390:{@[((1 0*+/^y)|^x)#0;!#x;:;x]}

x:3 3#1+!9

y:1 2 3 4

f390[x;y]

(1 2 3

4 5 6

7 8 9

0 0 0)

391. Conform table y columns to list y

x:4 2 # 9

y:5#8

a:((0 1*+/^y)|^x)#0

a

(0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0)

a[;!(^x)[1]]:x

a

(9 9 0 0 0

9 9 0 0 0

9 9 0 0 0

9 9 0 0 0)

392. matrix from scalar or vector

x:4

^x

!0

(1+~#^x),:/x

,,4

^(1+~#^x),:/x

1 1

x:7 8

^(1+~#^x),:/x:7 8

1 2

393. see 248

394. [deferred]

395. [deferred]

396. remove columns y from x

y

((1 2 3 4

5 6 7 8

9 10 11 12)

(13 14 15 16

17 18 19 20

21 22 23 24))

y _di\:\: 0 2

((2 4

6 8

10 12)

(14 16

18 20

22 24))

397. see 73

398. diagonals from columns

x:(1 2 3 4 5

6 7 8 9 10

11 12 13 14 15

16 17 18 19 20

21 22 23 24 25)

(-!5)!'x

(1 2 3 4 5

10 6 7 8 9

14 15 11 12 13

18 19 20 16 17

22 23 24 25 21)

399. columns from diagonals

x

(1 2 3 4 5

10 6 7 8 9

14 15 11 12 13

18 19 20 16 17

22 23 24 25 21)

(!5)!'x

(1 2 3 4 5

6 7 8 9 10

11 12 13 14 15

16 17 18 19 20

21 22 23 24 25)

400. [deferred]

401. first word in string x

x:"twas brillig and the slith"

x?" "

4

(x?" ")#x

"twas"

fw:{(x?" ")#x}

fw x

"twas"

402. see 392

403. [omitted]

404. end points for x fields of length y

x:5

y:3

@[&x*y;(y-1)+y*!x;:;1]

0 0 1 0 0 1 0 0 1 0 0 1 0 0 1

405. start points for x fields of length y

x:5

y:3

@[&x*y;y*!x;:;1]

1 0 0 1 0 0 1 0 0 1 0 0 1 0 0

406. add y to last item of x

x:1 2 3 4 5

y:100

@[x;-1+#x;+;y]

1 2 3 4 105

407. vector length y of x 1s, the rest 0s

x:5

y:12

@[&12;!x;:;1]

1 1 1 1 1 0 0 0 0 0 0 0

408. initial empty row to start matrix of x columns

x:15

,&x

,0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

,1.0*&x

,0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0

,x#" "

," "

409. [omitted]

410. number of columns in matrix x

x:2 7#" "

^x

2 7

*|^x

7

411. number of rows in matrix x

x:2 7#" "

^x

2 7

#x

2

412. see 184

413. omitted

414. ending indices from field lengths

x:4 7 13 15 20

(1+*x),-':x

5 3 6 2 5

415. lengths of infixes of 1 in x

x:0 0 1 1 1 0 0 1 1 1 1 0 1

a:<':0,x,0

b:>':0,x,0

a

0 0 0 0 0 1 0 0 0 0 0 1 0 1

b

0 0 1 0 0 0 0 1 0 0 0 0 1 0

&a

5 11 13

&b

2 7 12

(&a)-(&b)

3 4 1

(&<':0,x,0)-&>':0,x,0

3 4 1

m415:{(&<':0,x,0)-&>':0,x,0}

416. omitted

417. end points of equal infixes

x:"baackkkegtt"

(~(1 _ x)=-1 _ x),1

1 0 1 1 0 0 1 1 1 0 1

418. starting indices of equal item infix

x:"baackkkegtt"

1,~(1 _ x)=-1 _ x

1 1 0 1 1 0 0 1 1 1 0

419. pairwise ratios

x:2 10 50 100

%':x

5 5 2.0

420. see 383

421. deferred

422. see 154

423. lengths from start indicator

x:1 0 1 0 0 1 0 0 0 1 0

&x,1

0 2 5 9 11

-':&x,1

2 3 4 2

424. single blank from multiples

x:"a b c d"

x[&a|1 _ 1!1,a:~" "=x]

"a b c d"

425. see 380

426. change all multiple infixes of y in x to single

x:"bccbceekl"

y:"c"

x[&a|-1 _ 1,a:~x=y]

"bcbceekl"

427. deferred

428. see 73

429. matrix with diagonal x

x:5 9 6 7 2

(2##x)#,/x,'(2##x)#0

(5 0 0 0 0

0 9 0 0 0

0 0 6 0 0

0 0 0 7 0

0 0 0 0 2)

430. polynomial derivative

x:1 2 3 4 5

-1 _ x*|!#x

4 6 6 4

431. does item differ from next one

x:"ceefffmeksc"

(~=':x),1

1 0 1 0 0 1 1 1 1 1 1

432. does item differ from previous one

x:"ceefffmeksc"

1,~=':x

1 1 0 1 0 0 1 1 1 1 1

433. replace last item of x with y

x:"abbccdefcdab"

y:"t"

@[x;-1+#x;:;y]

"abbccdefcdat"

434. replace first item of x with y

x:"abbccdefcdab"

y:"t"

@[x;0;:;y]

"tbbccdefcdab"

435. [deferred]

436. [deferred]

x:"00002345600345000"

((x="0")?0)_ x

"2345600345000"

438. index of 1st 1 following index y in x

x:1 0 0 1 1 0 1 1 0

y:3

y+(y _ x)?1

3

y+((y+1)_ x)?1

3

(y+1)+((y+1)_ x)?1

4

a more K-like alternative

&x

0 3 4 6 7

y<&x

0 0 1 1 1

(&x)[*&y<&x]

4

439. last 1s in groups of 1s

x:0 1 1 0 1 1 1 0 0 1

x>1 _ x,0

0 0 1 0 0 0 1 0 0 1

440. 1st 1 in groups of 1s

x:0 1 1 0 1 1 1 0 0 1

x>1 _ x,0

0 0 1 0 0 0 1 0 0 1

441. comma separated list from table

x:("Swift";"Austen";"Dickens")

x

("Swift"

"Austen"

"Dickens")

",",'x

(",Swift"

",Austen"

",Dickens")

,/",",'x

",Swift,Austen,Dickens"

1 _ ,/",",'x

"Swift,Austen,Dickens"

442. 1st difference

x:+\1 2 3 4 5

x

1 3 6 10 15

-':x

2 3 4 5

(*x),(-':x)

1 2 3 4 5

443. drop first y columns from matrix x

x:4 3#!12

x

(0 1 2

3 4 5

6 7 8

9 10 11)

y:2

y _ x

(6 7 8

9 10 11)

444. drop first y rows from matrix x

x:4 3#!12

x

(0 1 2

3 4 5

6 7 8

9 10 11)

y:2

y _' x

(,2

,5

,8

,11)

445. number of columns in matrix x

x:4 3#!12

x

(0 1 2

3 4 5

6 7 8

9 10 11)

*|^x

3

446. number of rows in matrix x

x:4 3#!12

x

(0 1 2

3 4 5

6 7 8

9 10 11)

#x

4

447. conditional drop of y items from array x

x:4 3#!12

x

(0 1 2

3 4 5

6 7 8

9 10 11)

g:0

(y*g) _ x

(0 1 2

3 4 5

6 7 8

9 10 11)

g:1

(y*g) _ x

(6 7 8

9 10 11)

448. conditional drop of last item of array x

x:4 3#!12

x

(0 1 2

3 4 5

6 7 8

9 10 11)

y:0

(-y) _ x

(0 1 2

3 4 5

6 7 8

9 10 11)

y:1

(-y) _ x

(0 1 2

3 4 5

6 7 8)

449. limiting x between l and h, inclusive

x: 5 6 _draw 100

x

(58 9 37 84 39 99

60 30 45 97 77 35

49 87 82 79 8 30

46 61 20 51 12 34

31 51 29 35 17 89)

l:30

h:70

l|h&x

(58 30 37 70 39 70

60 30 45 70 70 35

49 70 70 70 30 30

46 61 30 51 30 34

31 51 30 35 30 70)

450. arithmetic precision of system in decimals

log10:{(_log x)%_log 10}

log10 3

0.47712125471966244

_ _abs log10 _abs 1-3*%3

0I

This indicates that the precision is infinite (which isn't true)

and is a consequence of

3*%3

1.0

451. arithmetic progression from x to y with step g

ap:{[x;y;g]x+g*!1+_(y-x)%g}

ap[3;20;5]

3 8 13 18

ap[3;-20;-5]

3 -2 -7 -12 -17

452. number of positions in integer x

dp:{1+(x<0)+_ log10[_abs[x+0=x]]}

dp 1234

4

dp -1234

5

dp 0

1

dp 1

1

dp 7

1

dp 12345678

8

453. round to nearest even integer

re:{_ x+~1>x!2}

x:0.9 1 2.5 3.1 -0.2 -1.9

re x

0 2 2 4 0 -2

454. rounding, but to nearest even integer if fractional part is 0.5

rn:{_ x+0.5*~0.5=x!2}

x:23.6 40.5 3.2 -14.02 3.5 4.5

rn x

24 40 3 -14 4 4

455. number of digit positions in x

nd:{1+_ log10(x=0)+x*1 -10[x<0]}

x:4 678 -21 -10854

nd x

1 3 3 6

456. number of digits in nonnegative integer x

np:{1+_ log10 x+0=x}

x:0 13 523 16008

np x

1 2 3 5

457. is x integral

ii:{x=_ x}

x:67 -120 3.83 -5.5

ii x

1 1 0 0

458. [omitted]

459. leading digit of numeric code abbb

ld:{_ x%1000}

x:6 _draw 10000

x

1319 8629 6581 6988 790 9045

ld x

1 8 6 6 0 9

460. round y to x decimals

rn:{(10^-x)*_ 0.5+y*10^x}

y:3.3256789

x:3

rn[x;y]

rn[x;y]

3.326

461. round to nearest hundredth

rh:{0.01*_0.5+x*100}

x:3.1414 2.71828 -12.66666

rh x

3.14 2.72 -12.67

462. round to nearest integer

ri:{_0.5+x}

x:4.5 3.21 80.9 -2.4 -9.6

ri x

5 3 81 -2 -10

463. is x a leap year

ly:{(+/0=x!/:4 100 400)!2}

x:1900 1901 1904 2000

ly x

0 0 1 1

464. framing character matrix x

x:4 4#"abcdefghijklmnop"

x

("abcd"

"efgh"

"ijkl"

"mnop")

+"-",'(+"|",'x,'"|"),'"-"

("------"

"|abcd|"

"|efgh|"

"|ijkl|"

"|mnop|"

"------")

465. magnitude of fractional part

x:6.13 -6.13

_abs[x]!1

0.13 0.13

466. remove every y-th item of x

x:4+!10

x

4 5 6 7 8 9 10 11 12 13

y:3

!#x

0 1 2 3 4 5 6 7 8 9

(!#x)!y

0 1 2 0 1 2 0 1 2 0

~0=(!#x)!y

0 1 1 0 1 1 0 1 1 0

&~0=(!#x)!y

1 2 4 5 7 8

x[&~0=(!#x)!y]

5 6 8 9 11 12

467. select every y-th item of y

x:4+!10

x

4 5 6 7 8 9 10 11 12 13

y:3

!#x

0 1 2 3 4 5 6 7 8 9

(!#x)!y

0 1 2 0 1 2 0 1 2 0

0=(!#x)!y

1 0 0 1 0 0 1 0 0 1

&0=(!#x)!y

0 3 6 9

x[&0=(!#x)!y]

4 7 10 13

468. see dv in 164

469. remove every second item

x:"abcdefghijklmn"

(!#x)!2

0 1 0 1 0 1 0 1 0 1 0 1 0 1

&(!#x)!2

1 3 5 7 9 11 13

x[&(!#x)!2]

"bdfhjln"

470. items of x divisible by y

x:10 _draw 100

x

95 33 64 10 78 1 47 20 92 95

y:4

x!y

3 1 0 2 2 1 3 0 0 3

0=x!y

0 0 1 0 0 0 0 1 1 0

&0=x!y

2 7 8

x[&0=x!y]

64 20 92

471. index of first occurrence of g in x (circularly) after y

x: 15 _draw 10

x

6 6 0 0 8 9 8 1 0 2 9 4 6 3 5

g:0 6 5

y:9

(y+(y!x)?g)!#x

9

(y+(y!x)?/:g)!#x

2 12 14

472. omitted

473. is x even

x:1 2 3 4 5

~x!2

0 1 0 1 0

474. round x to zero if magnitude less than y

x:1e-4 -1e-8 -1e-12 1e-16

x*~y>_abs x

0.0001 -1e-008 0 0

475. increase absolute value without sign change

x:0 -1 2 -3 4 -5

sg:{(x>0)-(x<0)}

y:10

(sg x)*y+_abs x /problem with x=0

0 -11 12 -13 14 -15.0

476. fractional part with sign

x:0.2 2.3 -0.2 -1.8 0 5 -7

(sg x)*(_abs x)!1

0.2 0.3 -0.2 -0.8 0 0 0

477. square x retaining sign

x:0 -1 2 -3 4

x*_abs x

0 -1 4 -9 16.0

478. fractional part

x:0 1 -2 3.4 -5.6 -6.1

x!1

0 0 0 0.4 0.4 0.9

479. last part of abbb

x:1234 5678 9012 345 6789

x!1000

234 678 12 345 789

480. replace items of x in y by 0

x:1 2 3 4 5

y:2 4

x _lin y

0 1 0 1 0

x*~x _lin y

1 0 3 0 5

481.replace items of x not in y by 0

x:1 2 3 4 5

y:2 4

x _lin y

0 1 0 1 0

x*x _lin y

0 2 0 4 0

482. merge x any under control of g

x:1 2 3 4 5

y:100 200 300 400 500

g:1 0 0 1 1 0 1 0 0 1

(x,y)[<<g]

100 1 2 200 300 3 400 4 5 500

483. see 481

484. right to left scan

x:1 2 3 4 5

|+\|x

15 14 12 9 5

485. append empty row on matrix

x:("ab";"cd";"ef")

x

("ab"

"cd"

"ef")

x,,(*|^x)#" "

("ab"

"cd"

"ef"

" ")

486. [omitted]

487. insert empty row in x after row y

x:("ab";"cd";"ef")

x

("ab"

"cd"

"ef")

y:1

a:x,,(*|^x)#" "

a

("ab"

"cd"

"ef"

" ")

b:<(!#x),y

b

0 1 3 2

a[b]

a[b]

("ab"

"cd"

" "

"ef")

(x,,(*|^x)#" ")[<(!#x),y]

("ab"

"cd"

" "

"ef")

488. omitted

489. make string y into table guided by marker x

y:"eachwordinarow"

x:1 0 0 0 1 0 0 0 1 0 1 1 0 0

a:&x

a

0 4 8 10 11

b:a _ y

b

("each"

"word"

"in"

,"a"

"row")

(&x)_ y

("each"

"word"

"in"

,"a"

"row")

490. insert spaces in text

x:"wider"

a:+,x

a

(,"w"

,"i"

,"d"

,"e"

,"r")

b:a,'" "

b

("w "

"i "

"d "

"e "

"r ")

,/b

"w i d e r "

,/(+,x),'" "

"w i d e r "

491. or-reduce infixes of y marked by x

x:1 0 0 1 0 0 0 1 0 0 0 0

y:0 0 0 0 1 0 0 0 0 0 1 0

a:(&x)_ y

a

(0 0 0

0 1 0 0

0 0 0 1 0)

|/'a

0 1 1

492. and-reduce infixes of y marked by x

x:1 0 0 1 0 0 0 1 0 0 0 0

y:0 1 1 1 1 1 1 1 1 1 1 0

(&x)_ y

(0 1 1

1 1 1 1

1 1 1 1 0)

&/'(&x)_ y

0 1 0

493. choose x or y depending on boolean g

x:"abcdef"

y:"xyz"

g:0

:[g;x;y]

"xyz"

g:1

:[g;x;y]

"abcdef"

494. see 424

495. indices of all occurrences of y in x

x:"abcdefgab"

y:"afc*"

x _lin y

1 0 1 0 0 1 0 1 0

&x _lin y

0 2 5 7

496. remove punctuation characters

x:"oh! no, stop it. you will?"

y:",;:.!?"

x _dvl y

"oh no stop it you will"

497. set union

x:"12345"

y:"4567890"

y,x[&~x _lin y]

"4567890123"

498. set difference

x:"12345"

y:"4567890"

x _dvl y

"123"

499. rows of y starting with item of x

x:("abcd";"efgh";"ijkl";"mnop")

x

("abcd"

"efgh"

"ijkl"

"mnop")

y:"ai"

x[;0] _lin y

1 0 1 0

&x[;0] _lin y

0 2

x[&x[;0] _lin y]

("abcd"

"ijkl")

500. set intersection

x:"abcdefghijxyz"

y:"yacqwopzbx"

x[&x _lin y]

"abcxyz"

501.see 154

502. deferred

503. indices of all occurrences of y in x

x:"abcdeabc"

y:"a"

&x=y

0 5

504. replace items of y satisfying x with g

x:1 0 0 0 1 0 1 1 0 1

y:"abcdefghij"

g:" "

@[y;&x;:;g]

" bcd f i "

505. see 154

506. see 41

507. insert blank in y after mark in x

x:1 0 0 0 0 1 0 0

y:"abcdefgh"

x#\:" "

(," "

""

""

""

""

," "

""

"")

y,' x#\:" "

("a "

,"b"

,"c"

,"d"

,"e"

"f "

,"g"

,"h")

,/ y,' x#\:" "

"a bcdef gh"

508. conditional text

x:0

,/((~x)#'"in"),"correct"

"incorrect"

x:1

,/((~x)#'"in"),"correct"

"correct"

509. remove y from x

x:"abcdeabc"

y:"a"

x _dv y

"bcdebc"

510. remove blanks

x:" bcde bc"

x _dv " "

"bcdebc"

511. apply f over all of x

x:2 3 4#1+!24

x

((1 2 3 4

5 6 7 8

9 10 11 12)

(13 14 15 16

17 18 19 20

21 22 23 24))

+//x

300

ao:{[f;x]f//x}

ao[+;x]

300

ao[*;1.0*x]

6.204484e+023

ao[+;-x]

-300

512. select items of x according to markers in y

x:2 3 4#1+!24

x

((1 2 3 4

5 6 7 8

9 10 11 12)

(13 14 15 16

17 18 19 20

21 22 23 24))

y:1 0 0 1

x[;;&y]

((1 4

5 8

9 12)

(13 16

17 20

21 24))

513. empty matrix

x:,!0

x

,!0

^x

1 0

514. apply to dimension 1 function defined on dimension 0

x:3 4#1+!12

x

(1 2 3 4

5 6 7 8

9 10 11 12)

+/x

15 18 21 24

+/'x

10 26 42

515. deferred

516. multiply each column of x by y

x:(1 2 3 4 5 6

7 8 9 10 11 12)

y:10 100

x*y

(10 20 30 40 50 60

700 800 900 1000 1100 1200)

y*x

(10 20 30 40 50 60

700 800 900 1000 1100 1200)

517. deferred

518. transpose matrix x on condition y

x:2 3#!6

x

(0 1 2

3 4 5)

y:1

y +:/x

(0 3

1 4

2 5)

y:0

y +:/x

(0 1 2

3 4 5)

519. see 85

520. see 86

521. matrix with x columns y

x:4

y:"abc"

x#'y

("aaaa"

"bbbb"

"cccc")

522. see 80

523. deferred

524. deferred

525. main diagonal

x:(1 2 3 4

5 6 7 8

9 10 11 12)

y:2#'!#x

y

(0 0

1 1

2 2)

x ./: y

1 6 11

526. see 525

527. transpose planes of three-dimensional x

x:2 2 2#!8

x

((0 1

2 3)

(4 5

6 7))

+:'x

((0 2

1 3)

(4 6

5 7))

528. vector (cross) product

x:2 8 5 6 3 1 7 7 10 4

y:6 9 1 1 6 7 1 4 1 5

((1!x)*-1!y)-(-1!x)*1!y

4 28 46 -27 -41 39 45 3 -19 -58

529. markers for x at y

x:"abcdefghijklmn"

y:3 7 9

@[&#x;y;:;1]

0 0 0 1 0 0 0 1 0 1 0 0 0 0

530. index of last occurrence of y in x

x:10 _draw 5

x

3 0 4 3 1 4 4 3 3 1

y:4

*|&x=y

6

y:3

*|&x=y

8

531. replace each item of y with index of its last occurrence

x:"aabbbcccc"

y:x,"ddd"

x

"aabbbcccc"

y

"aabbbccccddd"

(|x)?/:y

7 7 4 4 4 0 0 0 0 9 9 9

#x

9

(#x)-(|x)?/:y

2 2 5 5 5 9 9 9 9 0 0 0

0|-1+(#x)-(|x)?/:y

1 1 4 4 4 8 8 8 8 0 0 0

532. index of last occurrence of y in x, counted from the rear

x:8 4 9 1 5 7

y:8 2 3 4 9 5 7 1 10 6 8 2

(|x)?/:y

5 6 6 4 3 1 0 2 6 6 5 6

533. reverse x on condition y

x:1 2 3 4 5

y:0

y |:/x

1 2 3 4 5

y:1

y |:/x

5 4 3 2 1

534. see 203

535. avoiding parentheses using reverse

x:1 2 3 4 5

(#x),1

5 1

|1,#x

5 1

536. rotate rows left

x:3 4#1+!12

x

(1 2 3 4

5 6 7 8

9 10 11 12)

1!'x

(2 3 4 1

6 7 8 5

10 11 12 9)

537. rotate rows right

x:3 4#1+!12

x

(1 2 3 4

5 6 7 8

9 10 11 12)

-1!'x

(4 1 2 3

8 5 6 7

12 9 10 11)

538. insert 0 in list of ones x after indices y

x:1 1 1 1 1 1 1 1 1 1

y

1 3 7

+,x

(,1

,1

,1

,1

,1

,1

,1

,1

,1

,1)

@[+,x;y;{x,0}]

(,1

1 0

,1

1 0

,1

,1

,1

1 0

,1

,1)

,/@[+,x;y;{x,0}]

1 1 0 1 1 0 1 1 1 1 0 1 1

539. Boolean vector of length y with zeros in locations x

x:2 3 4 8

y:10

method A

~(!y) _lin x

1 1 0 0 0 1 1 1 0 1

method B

~@[&y;x;:;1]

1 1 0 0 0 1 1 1 0 1

540. markers in Boolean vector of length x at indices y

x:10

y:1 3 7

method A

(!x) _lin y

0 1 0 1 0 0 0 1 0 0

method B

@[&x;y;:;1]

0 1 0 1 0 0 0 1 0 0

541. omitted

542. omitted

543. see 540

544. do x and y match

x~y

545. zero items of y not in x

y: 2 3 4 5 6 7 8 9 10 11

x:2 3 5 7 11

y _lin x

1 1 0 1 0 1 0 0 0 1

y*y _lin x

2 3 0 5 0 7 0 0 0 11

546. is count of atoms 1 (uses cs from 366)

cs:({#,//x}

co:{1=cs[x]}

co[35]

1

co[,35]

1

co[1 1#35]

1

co[1 1 1#35]

1

co[1 2]

0

co[!0]

0

547. is x vector

iv:{1=#^x}

iv[0]

0

iv[1 2]

1

iv[(1;2)]

1

iv[(1 2;3 4)]

0

iv[2 3#!6]

0

iv[!0]

1

548. test if empty

ie:{0 _in ^x}

ie 7

0

ie 8 9

0

ie 2 3#6

0

ie (1 0#5)

1

549. alphabetic comparison (depends on storage values)

"a"<"b"

1

"a">"b"

0

_ic";"

59

_ic"/"

47

";"<"/"

0

550. see 37

551. index of first differing item of x and y

x:3 1 4 1 6 0

y:3 1 4 1 5 9

(~x=y)?1

4

552. which items of x are not in y

x:2 3 4 5 6 7 8 9 10 11

y:2 3 5 7 11

~x _lin y

0 0 1 0 1 0 1 1 1 0

553. see 37

554. select from g based on index of x in y

g:("William Shakespeare"

> "John Milton"

> "Jonathan Swift"

> "Jane Austen"

> "John Keats"

> "Charles Dickens")

y:1564 1608 1667 1775 1795 1812

x:1775

g[y?x]

"Jane Austen"

555. all axes of rectangular array x

x:2 2 2 2#!16

!#^x

0 1 2 3

556. all indices of vector x

x:2 2 2 2#!16

!#^x

0 1 2 3

557. arithmetic progression of y numbers from x with step g

x:5

y:8

g:100

x+g*!y

5 105 205 305 405 505 605 705

ap:{[x;g;y]x+g*!y}

ap[x;g;y]

5 105 205 305 405 505 605 705

558. consecutive integers from x to y

ci:{x+!1+y-x}

ci[5;10]

5 6 7 8 9 10

559. index of first marker in Boolean x

x:0 0 1 0 1 0 0 1 1 0

x?1

2

560. omitted

561. numeric code from character

x:" aA0"

_ic[x]

32 97 65 48

562. index of y in x

x:" abcdefgh"

x?/:y

6 1 4 5 4 0 8 5 1 4

x?/:y

4 5 1 6 0 1 4 4 5 9

563. empty vector

!0

!0

^!0

,0

""

""

^""

,0

564. is x within range ( y[0],y[1] )

ci:{(y[0]<x)&(x<y[1])}

y:3 8

x:5

ci[x;y]

1

x:3

ci[x;y]

0

x:2

ci[x;y]

0

x:8

ci[x;y]

0

x:9

ci[x;y]

0

565. is x within range [ y[0],y[1] ]

oi:{(~y[0]>x)&(~x>y[1])}

y:3 8

x:5

oi[x;y]

1

x:3

oi[x;y]

1

x:2

oi[x;y]

0

x:8

oi[x;y]

1

x:9

oi[x;y]

0

566. zero all items of Boolean x

x:0 1 0 1 1 0 0 1 1 1 0

0&x

0 0 0 0 0 0 0 0 0 0 0

567. select x or y depending on g

x:`hot `white `short `old

y:`cold `black `tall `young

g:1 0 0 1

x,'y

(`hot `cold

`white `black

`short `tall

`old `young)

(!#g),'g

(0 1

1 0

2 0

3 1)

(x,'y) ./: (!#g),'g

`cold `white `short `young

568. omitted

569. change y to one if x

y:10 5 7 12 20.0

x:0 1 0 1 1

y^~x

10 1 7 1 1.0

alternatively

@[y;&x;:;1]

10 1 7 1 1

570. x implies y

x:0 1 0 1

y:0 0 1 1

~x>y

1 0 1 1

571. x but not y

x:0 1 0 1

y:0 0 1 1

x>y

0 1 0 0

572. division by 0

dz:{(~0=x)*y%x+x=0}

y:10 15 -20

x:2 0 0

y%x

5 0i -0i

dz[x;y]

5 0 0.0

573. exclusive or

x:0 0 1 1

y:0 1 0 1

~x=y

0 1 1 0

574. y where x is 0

x:0 7 8 0 2

y:10 4 6 7 3

x+y*x=0

10 7 8 7 2

575. Kronecker delta of x and y

x:0 0 1 1

y:0 1 0 1

x=y

1 0 0 1

576. append y items g before each item of x

x:1 3 5

y:2

g:10

y#g

10 10

(y#g),/:x

(10 10 1

10 10 3

10 10 5)

,/(y#g),/:x

10 10 1 10 10 3 10 10 5

577. append y items g after each item of x

x:1 3 5

y:2

g:10

,/x,\:y#g

1 10 10 3 10 10 5 10 10

578. merge items from x and y alternately

x:1 3 5 7

y:2 4 6 8

,/x,'y

1 2 3 4 5 6 7 8

579. variable length lines

x:"by and by"

y:"God caught his eye"

,(x;y)

,("by and by"

"God caught his eye")

580. see 490

581. insert y after each item of x

x:"abc"

y:"d"

,/x,'y

582. see 197

583. array and its negative

x:1 -3 5

x,'-x

(1 -1

-3 3

5 -5)

584. omitted

585. omitted

586. omitted

587. first column as a matrix

x:(0 1 2 3

4 5 6 7

8 9 10 11)

x[;,0]

(,0

,4

,8)

588. 2-row matrix from two vectors

x:"abcd"

y:"efgh"

(,x),(,y)

("abcd"

"efgh")

589. 2-column matrix from two vectors

x:"abcd"

y:"efgh"

x,'y

("ae"

"bf"

"cg"

"dh")

590. increasing rank of y to rank of x

x:("abcd"

> "efgh")

x

("abcd"

"efgh")

y:"ijkl"

#^x

2

#^y

1

#^,y

2

((#^x)-#^y),:/y

,"ijkl"

591. reshape vector x into 2-column matrix

x:"abcdefgh"

((_ 0.5*#x),2)#x

("ab"

"cd"

"ef"

"gh")

592. vector from array

x:2 1 2 1 2 1#!8

^x

2 1 2 1 2 1

,//x

0 1 2 3 4 5 6 7

^,//x

,8

593. matrix of y rows, each x

x:"abcd"

y:3

y#,x

("abcd"

"abcd"

"abcd")

594. see 203

595. one-row matrix from vector

x:2 3 5 7 11

x

2 3 5 7 11

^x

,5

,x

,2 3 5 7 11

^,x

1 5

596. see 366

597. omitted

598. omitted

599. number of columns in array x

x:1 1 1 1 1 678#0

^x

1 1 1 1 1 678

*|^x

678

600. number of columns in matrix x

x:3 19#0

^x

3 19

(^x)[1]

19

601. number of rows in matrix x

x:17 2#0

#x

17

602. choosing according to sign

sg:{(x>0)-(x<0)}

sg -4.5 0 6.78

-1 0 1

y:"-0+"

x:-54

y[1+sg[x]]

"-"

x:0

y[1+sg[x]]

"0"

x:1234.5

y[1+sg[x]]

"+"

603. conditional change of sign

y:1+!6

y

1 2 3 4 5 6

x:0 1 0 1 1 0

y*1 -1[x]

1 -2 3 -4 -5 6

604. omitted

605. indexing plotting characters with Boolean index

x:~3 6 5 7 2<\:1+!7

x

(1 1 1 0 0 0 0

1 1 1 1 1 1 0

1 1 1 1 1 0 0

1 1 1 1 1 1 1

1 1 0 0 0 0 0)

" *"[x]

("*** "

"****** "

"***** "

"*******"

"** ")

606. omitted

607. vector from column of matrix

x:3 4#!12

x

(0 1 2 3

4 5 6 7

8 9 10 11)

x[;0]

0 4 8

608. zeroing an array

x:1 2 3 4

x

1 2 3 4

x[]:0

x

0 0 0 0

x:2 3#!6

x

(0 1 2

3 4 5)

x[;]:0

x

(0 0 0

0 0 0)

x:9

x

9

x:0

x

0

609. omitted

610. y cyclic repetitions of vector x

x:"abcd"

y:3

(y*#x)#x

"abcdabcdabcd"

611. multiply each row of x by vector y

x:3 4#1+!12

x

(1 2 3 4

5 6 7 8

9 10 11 12)

y:1 10 100 10000

x*\:y

(1 20 300 40000

5 60 700 80000

9 100 1100 120000)

612. rank of array y (number of dimensions)

x:2 1 2 1 3 1 4#0

#^x

7

#^9

0

#^7 8 9

1

#^(1 2 3;4 5 6)

2

613. see 441

614. array with shape of y and x as its rows

y:3 4#!12

y

(0 1 2 3

4 5 6 7

8 9 10 11)

x:"abcd"

(^y)#x

("abcd"

"abcd"

"abcd")

615. first atom in x

x:2 2 2 2#!8

x

(((0 1

2 3)

(4 5

6 7))

((0 1

2 3)

(4 5

6 7)))

*//x

0

616. scalar from one-item vector

x:,8

x

,8

""#x

8

x[0]

8

*x

8

617. omitted

618. deferred

619. deferred

620. omitted

621. see 583

622. retain value of items marked by y, zero others

x:3 7 15 1 292

y:1 0 1 1 0

x*y

3 0 15 1 0

623. conditional change of sign

x:-9

y:0

x*-1^y

-9.0

y:1

x*-1^y

9.0

624. zero numerical array

x:2 3#99

x

(99 99 99

99 99 99)

x*0

(0 0 0

0 0 0)

625. omitted

626. omitted

627. omitted

628. omitted

629. error to stop execution

&`

630. omitted

631. omitted

1001. payback

cumulative accumulation factors

(see Zark APL Tutor News 1998 Quarter 2)

caf:{[W;R]*\(#W)#1+R}

pay:{[B;T;R;W]C*B-+\W%(#W)#T _ 1,C:caf[W;R]}

The version pay2 replaces T by END, and permits END to be any value between 0 and 1.

pay2:{[B;END;R;W]CPA:*\A:1+R

CPA*B-+\W%CPA%A*1-END}

B initial balance

T time of withdrawal: 0 start of period, 1 end

R interest rate per period

W withdrawal amount

B:1000

T:0

R:0.05

W:200 300 400 200

pay[B;T;.R;W]

840 567 175.35 -25.8825

T:1

pay[B;T;.R;W]

850 592.5 222.125 33.23125

R:0.05 0.04 0.06 0.05

pay[B;T;R;W]

840 561.6 171.296 -30.1392

T:1

pay[B;T;R;W]

850 584 219.04 29.992

1002. round summands

ensure sum of rounded summands matches round of sum

f:{y%:x

i:_y

x*@[i;{(_.5+/x)#>x}[y-i];+;1]}

y:42.35 38.45 19.20

_.5++/y

100

_.5+y

42 38 19

+/_.5+y

99

x:1

z:f[x;y]

z

42 39 19

+/z

100

y:42.65 37.60 19.75

_.5++/y

100

_.5+y

43 38 20

+/_.5+y

101

z:f[x;y]

z

43 37 20

+/z

100

1003. maximum sum of infixes

f:{|/0(0|+)\x}

x:-100 2 3 4 -100 6 7 8 9 -100

f x

30

1004. range union

i:(1 3;8 10;11 12;2 4)

/ given ordered (lefts;rights)

/ interval 0 and where left is greater than 1+ max previous right

f:{(x i;1!y i:0,&x>1+y:-1!|\y)}

+f .+{x@<x}i /flip f apply flip sort i

(1 4;8 12)

1005. pointer chasing

For r a primitive root of prime p, the additive list formed by (r*!p)!p has an interesting property, first discussed by August Crelle in the early 19th century. For example, if we take such a list for the primitive root 3 of 7:

a:(3*!7)!7

/ list of successive sums of 3, starting from 0, mod 7:

a

0 3 6 2 5 1 4

then if we treat the items of this list as pointers, and write

a\1

1 3 2 6 4 5

we find that the new list is the successive powers of 3, mod 7.

1006. partitions of y with no part less than x

part:{t:x _!1+_ y%2

(,/t,''t _f'y-t),y}

part[3;10]

(3 3 4

3 7

4 6

5 5

10)

part[1;6]

(1 1 1 1 1 1

1 1 1 1 2

1 1 1 3

1 1 2 2

1 1 4

1 2 3

1 5

2 2 2

2 4

3 3

6)

#:'part[1]'1+!10

1 2 3 5 7 11 15 22 30 42

1007. Pascal's triangle

pt:{+':0,x,0}

4 pt\1

(1

1 1

1 2 1

1 3 3 1

1 4 6 4 1)

pt pt pt pt 1

1 4 6 4 1

4 pt/1

1 4 6 4 1

1008. Polygon area

area: 0.5*+/{-/y*|x}':

The dyad {-/y*|x} yields the determinant of a 2-by-2 matrix. The dyad area yields the area of a polygon whose x-y coordinates are, in order, the rows of a 2-column matrix, where the last row is the same as the first row.

x:(7 2;10 5;6 8;3 6;4 3;7 2)

area x

24.5

1009. Great circle distance

gcd:{_cos?(*/_sin x)+(*/_cos x)*_cos[-/y]}

The great circle distance in radians between two points on a sphere whose latitudes in radians are in x and longitudes in radians are in y.

nmr:{x*180*60%3.141592653589798238}

1011. Degrees from degrees and minutes

dfdm:{+/x%1 60}

1012. Fibonacci numbers

fn:{x{x,+/-2#x}/0 1}

fn:{x(|+\)\1 1.0}

1013. Tree from depth;value

tdv:{[d;v](1#v),(c _ d-1)_f'(c:&1=d)_ v}

1014. Depth from tree

dt:{0,/1+_f'1_ x}

1015. Value from tree

vt:{(1#x),/_f'1_ x}

d:0 1 2 2 1 1

v:0 1 2 3 4 5

t:tdv[d;v]

t

(0

(1

,2

,3)

,4

,5)

dt t

0 1 2 2 1 1

vt t

0 1 2 3 4 5

These 3 recursions stop when they run out of data