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u;.n Improved

initial writing: 2003-09-09 last updated: 2007-02-14

u;.n is now supported by special code for the following cases, for n= 1, 2, _1, or _2 and monad or dyad, unless otherwise indicated. Moreover, if ;@:g is supported by special code, then so is the equivalent [: ; g .

• the dyad ;@:(<;.0) when the right argument is a boolean or literal list
• f/;.n where f is atomic (= < <. + +. +: etc.)
• the result of u on each cut is boxed and atomic (for example if the leftmost verb in u is the monad <)
•  ;@:(<@u;.n)
• ;@:(<@(f/\);.n) and ;@:(<@(f/\.);.n) where f is atomic. Such cases compute the partitioned f-scan functions.

The following benchmarks demonstrate the improvements in time and space:

p=:  0=?1e5$10
i1=: ?1e5   $100
i2=: ?1e5  5$100
b1=: ?1e5   $2
b2=: ?1e5 12$2

ts=: 6!:2 , 7!:2@]

dtb=: #~ ([: +./\. ' '&~:)  NB. delete trailing blanks

lf=: 10{a.
s=: (?1e6$#t){t=: ((97+i.26){a.),26$' '
b=: ?1e6$2

ts=: 6!:2 , 7!:2@]

ts '      <@dtb;.1  x' [ x=: lf (0,?n$#s)}s
ts '([: ; <@dtb;.1) x'                      NB. delete trailing blanks
ts 'x#~(x=lf) ([: ; <@(+./\.);.1) '' ''~:x' NB. delete trailing blanks
ts 'p       <@(+./\.);.1  b' [ p=: (i.#b) e. 0,?n$#b
ts 'p ([: ; <@(+./\.);.1) b'

Expression		n		J 5.03		J 5.02		Ratio
p + /;.1 i1				0.0067520   6.694400e4		0.0373595   2.037440e6		5.53		30.43
p + /;.1 i2				0.0187937   2.635520e5		0.0582348   4.567680e6		3.10		17.33
p *./;.1 b1				0.0037164   1.779200e4		0.0343734   1.571008e6		9.25		88.30
p *./;.1 b2				0.0112738   1.324800e5		0.0467945   3.417088e6		4.15		25.79
<@dtb;.1 x
		20000		0.2964201   2.764800e6		0.4308346   6.992384e6		1.45		2.53
		40000		0.5552815   4.046464e6		0.8056859   1.097107e7		1.45		2.71
		60000		0.8143004   5.147328e6		1.1502487   1.454963e7		1.41		2.83
		80000		1.0695007   6.490112e6		1.5129828   1.858253e7		1.41		2.86
		100000		1.3321813   7.563072e6		1.8697742   2.205626e7		1.40		2.92
([: ; <@dtb;.1) x
		20000		0.2151028   1.052224e6		0.4813159   6.991488e6		2.24		6.64
		40000		0.3706147   1.051200e6		0.8320175   1.096083e7		2.24		10.43
		60000		0.5425854   1.050816e6		1.1930329   1.457338e7		2.20		13.87
		80000		0.7052606   1.050816e6		1.5473053   1.857261e7		2.19		17.67
		100000		0.8756591   1.050816e6		1.9243666   2.204339e7		2.20		20.98
x#~(x=lf) ([: ; <@(+./\.);.1) ' '~:x
		20000		0.0898821   4.197056e6		0.2540219   9.115904e6		2.83		2.17
		40000		0.0861936   4.197056e6		0.4006755   1.309920e7		4.65		3.12
		60000		0.0911792   4.197056e6		0.5452727   1.671565e7		5.98		3.98
		80000		0.0936711   4.197056e6		0.6927660   2.072307e7		7.40		4.94
		100000		0.0931079   4.197056e6		0.8457122   2.420698e7		9.08		5.77
p <@(+./\.);.1 b
		20000		0.1156180   2.787712e6		0.1610699   7.011968e6		1.39		2.52
		40000		0.2040379   4.075328e6		0.2891990   1.098374e7		1.42		2.70
		60000		0.2950473   5.207552e6		0.4189239   1.461914e7		1.42		2.81
		80000		0.3929801   6.546368e6		0.5521369   1.863699e7		1.40		2.85
		100000		0.4859380   7.610432e6		0.6947852   2.209325e7		1.43		2.90
p ([: ; <@(+./\.);.1) b
		20000		0.0284265   1.050432e6		0.2083843   7.016960e6		7.33		6.68
		40000		0.0283919   1.050432e6		0.3735461   1.100512e7		13.16		10.48
		60000		0.0304595   1.050432e6		0.5332460   1.460691e7		17.51		13.91
		80000		0.0302851   1.050432e6		0.6730847   1.864077e7		22.22		17.75
		100000		0.0335300   1.050432e6		0.7799402   2.210182e7		23.26		21.04

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