Power: Difference between revisions

From NARS2000
Jump to navigationJump to search
VisualWikitext
(Created page with "<table border="1" cellpadding="5" cellspacing="0" rules="none" summary=""> <tr> <td valign="top"><apll>Z←{L} <i>f</i/>⍣<i>g</i> R</apll></td> <td></td> <td></td> <...")
 
mNo edit summary
 
(12 intermediate revisions by the same user not shown)
Line 4: Line 4:
   <td></td>
   <td></td>
   <td></td>
   <td></td>
   <td>successively applies the function <apll><i>f</i></apll> (or, if there is a Left argument, the function <apll>L∘<i>f</i></apll>) to <apll>R</apll> until the expression <apll>Z<sub>n</sub> <i>g</i> Z<sub>n-1</sub></apll> returns a singleton <apll>1</apll>.  <apll>Z<sub>n</sub></apll> and <apll>Z<sub>n-1</sub></apll> represent two consecutive applications of <apll><i>f</i></apll> (or <apll>L∘<i>f</i></apll>) to <apll>R</apll> where <apll>Z<sub>n</sub> ←→ <i>f</i> Z<sub>n-1</sub></apll> (or <apll>Z<sub>n</sub> ←→ L∘<i>f</i> Z<sub>n-1</sub></apll>).</td>
   <td>successively applies the function <apll><i>f</i></apll> (or, if there is a Left argument, the function <apll>L∘<i>f</i></apll>) to <apll>R</apll> until the expression <apll>Z<sub>n</sub> <i>g</i> Z<sub>n-1</sub></apll> returns a singleton <apll>1</apll>.  <apll>Z<sub>n</sub></apll> and <apll>Z<sub>n-1</sub></apll> represent two consecutive applications of <apll><i>f</i></apll> (or <apll>L∘<i>f</i></apll>) to <apll>R</apll> where <apll>Z<sub>0</sub>←R</apll> and <apll>Z<sub>n</sub>←<i>f</i> Z<sub>n-1</sub></apll> (or <apll>Z<sub>n</sub>←L∘<i>f</i> Z<sub>n-1</sub></apll>).</td>
</tr>
</tr>


Line 11: Line 11:
   <td></td>
   <td></td>
   <td></td>
   <td></td>
   <td>for non-negative singleton integer <apll><i>b</i></apll>, successively applies the function <apll><i>f</i></apll> (or, if there is a Left argument, the function <apll>L∘<i>f</i></apll>) to <apll>R</apll>, <apll><i>b</i></apll> number of times.</td>
   <td>for non-negative integer scalar <apll><i>b</i></apll>, successively applies the function <apll><i>f</i></apll> (or, if there is a Left argument, the function <apll>L∘<i>f</i></apll>) to <apll>R</apll>, <apll><i>b</i></apll> number of times;<br/>
      for a negative integer scalar <apll><i>b</i></apll>, successively applies the inverse of the function <apll><i>f</i></apll> (or, if there is a Left argument, the inverse to the function <apll>L∘<i>f</i></apll>), <apll>|<i>b</i></apll> number of times</td>
</tr>
</tr>


<tr>
<tr>
   <td colspan="4"><apll>L</apll> and <apll>R</apll> are arbitrary arrays</td>
   <td colspan="4"><apll>L</apll> and <apll>R</apll> are arbitrary arrays.</td>
</tr>
</tr>


Line 23: Line 24:


<tr>
<tr>
   <td colspan="4">In the second case, <apll><i>b</i></apll> must be a non-negative integer singleton; otherwise a <apll>DOMAIN ERROR</apll> is signaled.</td>
   <td colspan="4">In the second case, <apll><i>b</i></apll> must be an integer scalar, otherwise a <apll>DOMAIN ERROR</apll> is signaled.</td>
</tr>
</tr>


Line 34: Line 35:
       sqrt 2
       sqrt 2
1.414213562373095
1.414213562373095
      sqrt 2<_x/>
1.41421356237309504880168872420969807857
      √2<_x/>
1.41421356237309504880168872420969807857
       fib←{⍵,+/¯2↑⍵} ⍝ Calculate a Fibonacci sequence
       fib←{⍵,+/¯2↑⍵} ⍝ Calculate a Fibonacci sequence
       fib⍣15 ⊢ 1 1
       fib⍣15   1 1
1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597  
1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597  
       pow←{⍵=0:,1 ⋄ ⍺+⍡×⍣(⍵-1) ⍺} ⍝ Raise a polynomial to a non-negative integer power
       pow←{⍵=0:,1 ⋄ ⍺+⍡×⍣(⍵-1) ⍺} ⍝ Raise a polynomial to a non-negative integer power
Line 43: Line 48:
       phi
       phi
1.618033988749894
1.618033988749894
      phi=0.5×1+√5
1
</pre></apll>
</pre></apll>
=Inverses=
<p>When the right operand to the Power Operator is a negative integer scalar, the inverse of the function left operand is applied to the right argument.  If <apll>f f⍣¯1 R ←→ R</apll>, then <apll>f⍣¯1</apll> is a Right Identity Function, and if <apll>f⍣¯1 f R ←→ R</apll>, then <apll>f⍣¯1</apll> is a Left Identity Function, which all of the examples below satisfy.</p>
<p>At the moment, only a few inverse functions are available as follows:</p>
<table border="1" cellpadding="5" cellspacing="0" summary="">
<tr>
  <td>'''Function'''</td>
  <td>'''Meaning of Inverse'''</td>
  <td>'''Left/Right<br>Identity<br />Function'''</td>
</tr>
<tr>
  <td valign="top"><apll>    L⊥⍣¯1  R</apll></td>
  <td><apll><apll>(N⍴L)⊤R</apll> for <apll>N</apll> sufficiently large to display all digits of <apll>R</apll><br />
<div style="display: inline;"><apll>        10⊥⍣¯1      1234567890<br />1 2 3 4 5 6 7 8 9 0<br />      10⊥10⊥⍣¯1      1234567890<br />1234567890<br />        10⊥⍣¯1  10⊥1 2 3 4 5 6 7 8 9 0<br />1 2 3 4 5 6 7 8 9 0 </apll></div></td>
<td>L and R</td>
</tr>
<tr>
  <td valign="top"><apll>      ⊥⍣¯1  R</apll></td>
  <td><apll>2⊥⍣¯1  R</apll><br />
<div style="display: inline;"><apll>        ⊥⍣¯1  19<br />1 0 0 1 1<br />      2⊥⊥⍣¯1  19<br />19<br />        ⊥⍣¯1  2⊥1 0 0 1 1<br />1 0 0 1 1 </apll></div></td>
<td>L and R</td>
</tr>
<tr>
  <td valign="top"><apll>    L⊤⍣¯1  R</apll></td>
  <td><apll>L⊥R</apll><br />
<div style="display: inline;"><apll>              10 10 10⊤⍣¯1            2 3 4<br />234<br />      10 10 10⊤10 10 10⊤⍣¯1            2 3 4<br />2 3 4<br />              10 10 10⊤⍣¯1  10 10 10⊤234<br />234</apll></div></td>
<td>L and R</td>
</tr>
<tr>
  <td valign="top"><apll>    ×/⍣¯1  R</apll></td>
  <td><apll>πR</apll>, that is, factor <apll>R</apll> into primes<br />
<div style="display: inline;"><apll>        ×/⍣¯1  130<br />2 5 13<br />      ×/×/⍣¯1  130<br />130<br />        ×/⍣¯1 ×/2 5 13<br />2 5 13</apll></div></td>
<td>L and R</td>
</tr>
<tr>
  <td valign="top"><apll>      π⍣¯1  R</apll></td>
  <td><apll>×/R</apll>, that is, multiply together the prime factors in <apll>R</apll><br />
<div style="display: inline;"><apll>      π⍣¯1  2 5 13<br />130<br />      ππ⍣¯1  2 5 13<br />2 5 13<br />      π⍣¯1 π 130<br />130</apll></div></td>
<td>L and R</td>
</tr>
<tr>
  <td valign="top"><apll>  +\[X]⍣¯1  R</apll></td>
  <td><apll>¯2-\[X] R</apll><br />
<div style="display: inline;"><apll>        +\⍣¯1  ⍳4<br />1 1 1 1<br />      +\+\⍣¯1  ⍳4<br />1 2 3 4<br />        +\⍣¯1+\ ⍳4<br />1 2 3 4</apll></div></td>
<td>L and R</td>
</tr>
<tr>
  <td valign="top"><apll>¯2-\[X]⍣¯1  R</apll></td>
  <td><apll>+\[X] R</apll><br />
<div style="display: inline;"><apll>          ¯2-\⍣¯1        4⍴1<br />1 2 3 4<br />      ¯2-\¯2-\⍣¯1        4⍴1<br />1 1 1 1<br />          ¯2-\⍣¯1    ¯2-\4⍴1 <br />1 1 1 1</apll></div></td>
<td>L and R</td>
</tr>
<tr>
  <td valign="top"><apll>  -\[X]⍣¯1  R</apll></td>
  <td><apll>(2-\[X] R)×[X](⍴R)[X]⍴¯1 1</apll><br />
<div style="display: inline;"><apll>      a←?5⍴10<br />      a<br />10 8 3 2 6<br />        -\⍣¯1  a<br />10 2 ¯5 1 4<br />      -\-\⍣¯1  a<br />10 8 3 2 6<br />        -\⍣¯1-\ a<br />10 8 3 2 6 </apll></div></td>
<td>L and R</td>
</tr>
<tr>
  <td valign="top"><apll>  ÷\[X]⍣¯1  R</apll></td>
  <td><apll>(2÷\[X] R)*[X](⍴R)[X]⍴¯1 1</apll><br />
<div style="display: inline;"><apll>      a←?5⍴10<br />      a<br />10 8 3 2 6<br />        ÷\⍣¯1  a<br />10 1.25 0.375 1.5 3<br />      ÷\÷\⍣¯1  a<br />10 8 3 2 6<br />        ÷\⍣¯1÷\ a<br />10 8 3 2 6</apll></div></td>
<td>L and R</td>
</tr>
<tr>
  <td valign="top"><apll>  +∘÷/⍣¯1  R</apll></td>
  <td>Display the [https://en.wikipedia.org/wiki/Continued_fraction Continued Fraction] expansion of <apll>R</apll> to at most <apll>⎕PP</apll> terms<br />This function is best used on Multiple Precision numbers<br />
<div style="display: inline;"><apll>          +∘÷/⍣¯1    449<_r/>303<br />1 2 13 3 1 2<br />      +∘÷/+∘÷/⍣¯1    449<_r/>303<br />449<_r/>303<br />          +∘÷/⍣¯1 +∘÷/1 2 13 3 1 2<_x/><br />1 2 13 3 1 2</apll></div></td>
<td>L and R</td>
</tr>
<tr>
  <td valign="top"><apll>  L+∘÷/⍣¯1  R</apll></td>
  <td>Display the [https://en.wikipedia.org/wiki/Continued_fraction Continued Fraction] expansion of <apll>R</apll> to at most <apll>L</apll> terms<br />This function is best used on Multiple Precision numbers<br />
<div style="display: inline;"><apll>      phi←1+∘÷⍣=1  ⍝ Phi -- the Golden Ratio<br />      20 +∘÷/⍣¯1  phi<br />1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1<br />      20 +∘÷/⍣¯1  √2<br />1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2<br />      20 +∘÷/⍣¯1  *1<_x/><br />2 1 2 1 1 4 1 1 6 1 1 8 1 1 10 1 1 12 1 1<br />      20 +∘÷/⍣¯1  ○1<_x/><br />3 7 15 1 292 1 1 1 2 1 3 1 14 2 1 1 2 2 2 2<br />      +∘÷/35 +∘÷/⍣¯1  ○1<_x/><br />3.141592653589793238462643383279502884197<br />      ○1<_x/><br />3.141592653589793238462643383279502884197<br />          4 +∘÷/⍣¯1 +∘÷/○1<_x/><br />3 7 15 1<br />      +∘÷\4 +∘÷/⍣¯1 'r' ⎕DC ○1<_x/>    ⍝ Convergents to Pi<br />3 22<_r/>7 333<_r/>106 355<_r/>113</apll></div></td>
<td>L and R</td>
</tr>
</table>

Latest revision as of 17:17, 12 February 2024

Z←{L} f⍣g R successively applies the function f (or, if there is a Left argument, the function L∘f) to R until the expression Zn g Zn-1 returns a singleton 1. Zn and Zn-1 represent two consecutive applications of f (or L∘f) to R where Z0←R and Zn←f Zn-1 (or Zn←L∘f Zn-1).
Z←{L} f⍣b R for non-negative integer scalar b, successively applies the function f (or, if there is a Left argument, the function L∘f) to R, b number of times;
for a negative integer scalar b, successively applies the inverse of the function f (or, if there is a Left argument, the inverse to the function L∘f), |b number of times
L and R are arbitrary arrays.
In the first case, Zn g Zn-1 must return a Boolean-valued singleton; otherwise a DOMAIN ERROR is signaled.
In the second case, b must be an integer scalar, otherwise a DOMAIN ERROR is signaled.


For example,

      sqrt←{{0.5×⍵+⍺÷⍵}⍣=⍨⍵} ⍝ Calculate square root using Newton's method
      sqrt 2
1.414213562373095
      sqrt 2x
1.41421356237309504880168872420969807857
      √2x
1.41421356237309504880168872420969807857
      fib←{⍵,+/¯2↑⍵} ⍝ Calculate a Fibonacci sequence
      fib⍣15   1 1
1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 
      pow←{⍵=0:,1 ⋄ ⍺+⍡×⍣(⍵-1) ⍺} ⍝ Raise a polynomial to a non-negative integer power
      1 ¯1 pow 5
1 ¯5 10 ¯10 5 ¯1
      phi←1+∘÷⍣=1 ⍝ Calculate the Golden Ratio
      phi
1.618033988749894
      phi=0.5×1+√5
1

Inverses

When the right operand to the Power Operator is a negative integer scalar, the inverse of the function left operand is applied to the right argument. If f f⍣¯1 R ←→ R, then f⍣¯1 is a Right Identity Function, and if f⍣¯1 f R ←→ R, then f⍣¯1 is a Left Identity Function, which all of the examples below satisfy.

At the moment, only a few inverse functions are available as follows:

Function Meaning of Inverse Left/Right
Identity
Function
L⊥⍣¯1 R (N⍴L)⊤R for N sufficiently large to display all digits of R
10⊥⍣¯1 1234567890
1 2 3 4 5 6 7 8 9 0
10⊥10⊥⍣¯1 1234567890
1234567890
10⊥⍣¯1 10⊥1 2 3 4 5 6 7 8 9 0
1 2 3 4 5 6 7 8 9 0
L and R
⊥⍣¯1 R 2⊥⍣¯1 R
⊥⍣¯1 19
1 0 0 1 1
2⊥⊥⍣¯1 19
19
⊥⍣¯1 2⊥1 0 0 1 1
1 0 0 1 1
L and R
L⊤⍣¯1 R L⊥R
10 10 10⊤⍣¯1 2 3 4
234
10 10 10⊤10 10 10⊤⍣¯1 2 3 4
2 3 4
10 10 10⊤⍣¯1 10 10 10⊤234
234
 L and R
×/⍣¯1 R πR, that is, factor R into primes
×/⍣¯1 130
2 5 13
×/×/⍣¯1 130
130
×/⍣¯1 ×/2 5 13
2 5 13
 L and R
π⍣¯1 R ×/R, that is, multiply together the prime factors in R
π⍣¯1 2 5 13
130
ππ⍣¯1 2 5 13
2 5 13
π⍣¯1 π 130
130
 L and R
+\[X]⍣¯1 R ¯2-\[X] R
+\⍣¯1 ⍳4
1 1 1 1
+\+\⍣¯1 ⍳4
1 2 3 4
+\⍣¯1+\ ⍳4
1 2 3 4
 L and R
¯2-\[X]⍣¯1 R +\[X] R
¯2-\⍣¯1 4⍴1
1 2 3 4
¯2-\¯2-\⍣¯1 4⍴1
1 1 1 1
¯2-\⍣¯1 ¯2-\4⍴1
1 1 1 1
 L and R
-\[X]⍣¯1 R (2-\[X] R)×[X](⍴R)[X]⍴¯1 1
a←?5⍴10
a
10 8 3 2 6
-\⍣¯1 a
10 2 ¯5 1 4
-\-\⍣¯1 a
10 8 3 2 6
-\⍣¯1-\ a
10 8 3 2 6
L and R
÷\[X]⍣¯1 R (2÷\[X] R)*[X](⍴R)[X]⍴¯1 1
a←?5⍴10
a
10 8 3 2 6
÷\⍣¯1 a
10 1.25 0.375 1.5 3
÷\÷\⍣¯1 a
10 8 3 2 6
÷\⍣¯1÷\ a
10 8 3 2 6
 L and R
+∘÷/⍣¯1 R Display the Continued Fraction expansion of R to at most ⎕PP terms
This function is best used on Multiple Precision numbers
+∘÷/⍣¯1 449r303
1 2 13 3 1 2
+∘÷/+∘÷/⍣¯1 449r303
449r303
+∘÷/⍣¯1 +∘÷/1 2 13 3 1 2x
1 2 13 3 1 2
 L and R
L+∘÷/⍣¯1 R Display the Continued Fraction expansion of R to at most L terms
This function is best used on Multiple Precision numbers
phi←1+∘÷⍣=1 ⍝ Phi -- the Golden Ratio
20 +∘÷/⍣¯1 phi
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
20 +∘÷/⍣¯1 √2
1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
20 +∘÷/⍣¯1 *1x
2 1 2 1 1 4 1 1 6 1 1 8 1 1 10 1 1 12 1 1
20 +∘÷/⍣¯1 ○1x
3 7 15 1 292 1 1 1 2 1 3 1 14 2 1 1 2 2 2 2
+∘÷/35 +∘÷/⍣¯1 ○1x
3.141592653589793238462643383279502884197
○1x
3.141592653589793238462643383279502884197
4 +∘÷/⍣¯1 +∘÷/○1x
3 7 15 1
+∘÷\4 +∘÷/⍣¯1 'r' ⎕DC ○1x ⍝ Convergents to Pi
3 22r7 333r106 355r113
 L and R
Retrieved from "https://wiki.nars2000.org/index.php?title=Power&oldid=3871"