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11A. Inverse
J provides a comprehensive calculus of inverses:
| a0=: I=: ^:_1 | Inverse (adverb) |
| m1 =: ^ I | Natural log (^.); Inverse exponential |
| m2 =: ^.I | Exponential |
| m3 =: 10&^. | Base 10 log |
| m4 =: m3 I | Inverse base 10 log (10&^) |
| m5 =: ] -: m4@m3 | Tautology (test that m4 is left inverse) |
| m6 =: ] -: m3@m4 | Tautology (test that m4 is right inverse) |
| m7 =: ssc=: +/\ | Sum scan (subtotals) |
| m8 =: ssc I | Inverse sum scan; first differences |
| m9 =: (]-:m8@m7)*.(]-:m7@m8) | Tautology |
| m10=: assc=: -/\ | Alternating sum scan |
| m11=: assc I | |
| m12=: ] -: | |
| m13=: * /\ I | e.g. * /\ m13 3 1 4 15 9 26 5 3 |
| m14=: % /\ I | e.g. % /\ m14 3 1 4 15 9 26 5 3 |
| m15=: ~:/\ I | e.g. ~:/\ m15 1 1 0 1 1 0 1 1 |
| m16=: = /\ I | e.g. = /\ m16 1 1 0 1 1 0 1 1 |
| m17=: + /\. I | e.g. + /\. m17 3 1 4 15 9 26 5 3 |
| m18=: - /\. I | e.g. - /\. m18 3 1 4 15 9 26 5 3 |
| m19=: * /\. I | e.g. * /\. m19 3 1 4 15 9 26 5 3 |
| m20=: % /\. I | e.g. % /\. m20 3 1 4 15 9 26 5 3 |
| m21=: ~:/\. I | e.g. ~:/\. m21 1 1 0 1 1 0 1 1 |
| m22=: = /\. I | e.g. =/\. m22 1 1 0 1 1 0 1 1 |
| d23=: # I | Expand; 'ab' -: 1 0 1 # 1 0 1 d23 'ab' |
| m24=: p: I | p (n) the number of primes less than n |
| m25=: x: I | Floating point approx. of a rational. e.g. m25 3r7 |
| m26=: 1&+ I | Inverse increment; decrement |
| m27=: +&1 I | Inverse increment; decrement |
| m28=: >: I | Inverse increment; decrement |
| m29=: _1&+ I | Inverse decrement; increment |
| m30=: +&_1 I | Inverse decrement; increment |
| m31=: -&1 I | Inverse decrement; increment |
| m32=: <: I | Inverse decrement; increment |
| m33=: 2&* I | Inverse double; halve |
| m34=: *&2 I | Inverse double; halve |
| m35=: +: I | Inverse double; halve |
| m36=: 0.5&* I | Inverse halve; double |
| m37=: *&0.5 I | Inverse halve; double |
| m38=: %&2 I | Inverse halve; double |
| m39=: -: I | Inverse halve; double |
| m40=: ^&2 I | Inverse square |
| m41=: ^&3 I | Inverse cube |
| m42=: ^&0.5 I | Inverse square root |
| m43=: ^&1r3 I | Inverse cube root |
| m44=: 2&^ I | Inverse 2 with power; base 2 log |
| m45=: 10&^ I | Inverse 10 with power; base 10 log |
| m46=: 2&! I | Inverse triangular number. e.g. +/i.<.2&! I m |
| m47=: +~ I | Inverse double |
| m48=: *~ I | Inverse square |
| m49=: ^~ I | e.g. x^x=: ^~ I 12 |
| m50=: (3&+)@(%&2)I -: (%&2 I)@(3&+ I) | Inverse of composition is composition of inverses |
These inverses may be illustrated as follows:
x=: 2 3 5 7 ,.(] ; m1 ; m2 ; m1@m2) x +---------------------------------+ ¦2 3 5 7 ¦ +---------------------------------¦ ¦0.6931472 1.09861 1.60944 1.94591¦ +---------------------------------¦ ¦7.38906 20.0855 148.413 1096.63 ¦ +---------------------------------¦ ¦2 3 5 7 ¦ +---------------------------------+ (] ; m7 ; m8 ; m9) x +---------------------------+ ¦2 3 5 7¦2 5 10 17¦2 1 2 2¦1¦ +---------------------------+ (];m10;m11;m11@m10) x +-----------------------------------+ ¦2 3 5 7¦2 _1 4 _3¦2 _1 2 _2¦2 3 5 7¦ +-----------------------------------+