Condense: Difference between revisions

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     <table border="0" cellpadding="5" cellspacing="0" summary="">
     <table border="0" cellpadding="5" cellspacing="0" summary="">
     <tr>
     <tr>
       <td valign="top"><apll>Z←&lt;R</apll></td>
       <td valign="top"><apll>Z←&lt;R</apll> &nbsp;&nbsp; or &nbsp;&nbsp; <apll>Z←&lt;[X] R</apll></td>
       <td></td>
       <td></td>
       <td></td>
       <td></td>
       <td>converts <apll>R</apll> to a Hypercomplex array if <apll>(≢R)∊2 4 8</apll> or a Real array if <apll>1=≢R</apll>.</td>
       <td>converts <apll>R</apll> to a Hypercomplex array if <apll>(⍴R)[X]∊2 4 8</apll> or a Real array if <apll>(⍴R)[X]=1</apll>.</td>
     </tr>
     </tr>
     </table>
     </table>
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</tr>
</tr>
<tr>
<tr>
   <td><apll>R</apll> is an arbitrary Real numeric array (BOOL, INT, FLT, APA, RAT, VFP &mdash; otherwise, <apll>DOMAIN ERROR</apll>) whose number of columns (<apll>≢R</apll>) is <apll>1</apll>, <apll>2</apll>, <apll>4</apll>, or <apll>8</apll> &mdash; otherwise, <apll>LENGTH ERROR</apll>.</td>
   <td><apll>X</apll> is an optional numeric singleton axis with <apll>X∊⍳⍴⍴R</apll> &mddash; otherwise (including if <apll>R</apll> is a scalar), an <apll>AXIS ERROR</apll> is signalled.  If <apll>[X]</apll> is omitted, the operation applies to the last axis.</td>
</tr>
</tr>
<tr>
<tr>
   <td><apll>Z</apll> is the corresponding Real or Hypercomplex array of shape <apll>¯1↓⍴R</apll> using the columns of <apll>R</apll> as the coefficients of the resulting Real or Hypercomplex array.  If <apll>≢R</apll> is <apll>1</apll>, the result is the Real array <apll>(¯1↓⍴R)⍴R</apll>, if <apll>≢R</apll> is <apll>2</apll>, the result is a Complex array, if <apll>≢R</apll> is <apll>4</apll>, the result is a Quaternion array, and if <apll>≢R</apll> is <apll>8</apll>, the result is an Octonion array.</td>
  <td><apll>R</apll> is an arbitrary Real numeric array, i.e., Fixed-Precision Boolean, Integer, Floating Point, Arithmetic Progression Array, or Multiple-Precision Integer/Rational or Floating Point &mdash; otherwise, a <apll>DOMAIN ERROR</apll> is signalled.  The <apll>X</apll>-axis length <apll>(⍴R)[X]</apll> must be <apll>1</apll>, <apll>2</apll>, <apll>4</apll>, or <apll>8</apll> &mdash; otherwise, a <apll>LENGTH ERROR</apll> is signalled.</td>
</tr>
<tr>
   <td><apll>Z</apll> is the corresponding Real or Hypercomplex array of shape <apll>(X≠⍳⍴⍴R)/⍴R</apll> using the numbers along the <apll>X</apll>-axis of <apll>R</apll> as the coefficients of the resulting Real or Hypercomplex array.  If
<ul>
  <li><apll>(⍴R)[X]</apll> is <apll>1</apll>, the result is an array of Real numbers,</li>
  <li><apll>(⍴R)[X]</apll> is <apll>2</apll>, the result is an array of Complex numbers,</li>
  <li><apll>(⍴R)[X]</apll> is <apll>4</apll>, the result is an array of Quaternion numbers,</li>
  <li><apll>(⍴R)[X]</apll> is <apll>8</apll>, the result is an array of Octonion numbers.</li>
</ul>
</td>
</tr>
</tr>
</table>
</table>
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<p>For example,</p>
<p>For example,</p>


<apll>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;23<br />
<apll>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;23
23<br />
23
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;10 20<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;10 20
10<pn>J</pn>20<br />
10<pn>J</pn>20
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;2 4⍴⍳8<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;2 4⍴⍳8
1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4 5<pn>i</pn>6<pn>j</pn>7<pn>k</pn>8<br />
1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4 5<pn>i</pn>6<pn>j</pn>7<pn>k</pn>8
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;2 8⍴(⍳8),⌽⍳8<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;[1] 2 4⍴⍳8
1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4<pn>l</pn>5<pn>ij</pn>6<pn>jk</pn>7<pn>kl</pn>8 8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5<pn>l</pn>4<pn>ij</pn>3<pn>jk</pn>2<pn>kl</pn>1<br />
1<pn>J</pn>5 2<pn>J</pn>6 3<pn>J</pn>7 4<pn>J</pn>8
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 1⍴⍳6<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;2 8⍴(⍳8),⌽⍳8
1 2 3<br />
1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4<pn>l</pn>5<pn>ij</pn>6<pn>jk</pn>7<pn>kl</pn>8 8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5<pn>l</pn>4<pn>ij</pn>3<pn>jk</pn>2<pn>kl</pn>1
4 5 6<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;[1] 2 8⍴(⍳8),⌽⍳8
2 3<br />
1<pn>J</pn>8 2<pn>J</pn>7 3<pn>J</pn>6 4<pn>J</pn>5 5<pn>J</pn>4 6<pn>J</pn>3 7<pn>J</pn>2 8<pn>J</pn>1
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 2⍴(⍳6),⌽⍳6<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 1⍴⍳6
1<pn>J</pn>2&nbsp;3<pn>J</pn>4&nbsp;5<pn>J</pn>6<br />
1 2 3
6<pn>J</pn>5&nbsp;4<pn>J</pn>3&nbsp;2<pn>J</pn>1<br />
4 5 6
2 3<br />
2 3
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 4⍴(⍳12),⌽⍳12<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 2⍴(⍳6),⌽⍳6
1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4&nbsp;&nbsp;&nbsp;&nbsp;5<pn>i</pn>6<pn>j</pn>7<pn>k</pn>8&nbsp;9<pn>i</pn>10<pn>j</pn>11<pn>k</pn>12<br />
1<pn>J</pn>2&nbsp;3<pn>J</pn>4&nbsp;5<pn>J</pn>6
12<pn>i</pn>11<pn>j</pn>10<pn>k</pn>9&nbsp;8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5&nbsp;4<pn>i</pn>3<pn>j</pn>2<pn>k</pn>1<br />
6<pn>J</pn>5&nbsp;4<pn>J</pn>3&nbsp;2<pn>J</pn>1
2 3<br />
2 3
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 8⍴(⍳24),⌽⍳24<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;[1] 2 3 2⍴(⍳6),⌽⍳6
&nbsp;1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4<pn>l</pn>5<pn>ij</pn>6<pn>jk</pn>7<pn>kl</pn>8&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;9<pn>i</pn>10<pn>j</pn>11<pn>k</pn>12<pn>l</pn>13<pn>ij</pn>14<pn>jk</pn>15<pn>kl</pn>16&nbsp;17<pn>i</pn>18<pn>j</pn>19<pn>k</pn>20<pn>l</pn>21<pn>ij</pn>22<pn>jk</pn>23<pn>kl</pn>24<br />
1<pn>J</pn>6 2<pn>J</pn>5
24<pn>i</pn>23<pn>j</pn>22<pn>k</pn>21<pn>l</pn>20<pn>ij</pn>19<pn>jk</pn>18<pn>kl</pn>17&nbsp;16<pn>i</pn>15<pn>j</pn>14<pn>k</pn>13<pn>l</pn>12<pn>ij</pn>11<pn>jk</pn>10<pn>kl</pn>9&nbsp;&nbsp;&nbsp;8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5<pn>l</pn>4<pn>ij</pn>3<pn>jk</pn>2<pn>kl</pn>1<br />
3<pn>J</pn>4 4<pn>J</pn>3
2 3<br />
5<pn>J</pn>2 6<pn>J</pn>1
3 2
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 4⍴(⍳12),⌽⍳12
&nbsp;1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4&nbsp;&nbsp;&nbsp;5<pn>i</pn>6<pn>j</pn>7<pn>k</pn>8&nbsp;9<pn>i</pn>10<pn>j</pn>11<pn>k</pn>12
12<pn>i</pn>11<pn>j</pn>10<pn>k</pn>9&nbsp;8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5&nbsp;4<pn>i</pn>3<pn>j</pn>2<pn>k</pn>1
2 3
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;⍴⎕←&lt;2 3 8⍴(⍳24),⌽⍳24
&nbsp;1<pn>i</pn>2<pn>j</pn>3<pn>k</pn>4<pn>l</pn>5<pn>ij</pn>6<pn>jk</pn>7<pn>kl</pn>8&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;9<pn>i</pn>10<pn>j</pn>11<pn>k</pn>12<pn>l</pn>13<pn>ij</pn>14<pn>jk</pn>15<pn>kl</pn>16&nbsp;17<pn>i</pn>18<pn>j</pn>19<pn>k</pn>20<pn>l</pn>21<pn>ij</pn>22<pn>jk</pn>23<pn>kl</pn>24
24<pn>i</pn>23<pn>j</pn>22<pn>k</pn>21<pn>l</pn>20<pn>ij</pn>19<pn>jk</pn>18<pn>kl</pn>17&nbsp;16<pn>i</pn>15<pn>j</pn>14<pn>k</pn>13<pn>l</pn>12<pn>ij</pn>11<pn>jk</pn>10<pn>kl</pn>9&nbsp;&nbsp;&nbsp;8<pn>i</pn>7<pn>j</pn>6<pn>k</pn>5<pn>l</pn>4<pn>ij</pn>3<pn>jk</pn>2<pn>kl</pn>1
2 3
</apll>
</apll>


==Identities==
==Identities==


<apll>R ←→ &lt;&gt;R</apll> &nbsp;&nbsp;for all <apll>R</apll> (see [[Dilate]] for the definition of monadic Right Caret)<br />
<apll> R ←→ &lt;   &gt;   R</apll> &nbsp;&nbsp;for all Real (<apll>1==R</apll>) <apll>R</apll> (see [[Dilate]] for the definition of monadic Right Caret)<br />
<apll>R ←→ &gt;&lt;R</apll> &nbsp;&nbsp;for all <apll>R</apll> with <apll>(¯1↑⍴R)∊1 2 4 8</apll><br />
<apll> R ←→ &lt;[X] &gt;[X] R</apll> &nbsp;&nbsp;for all Real (<apll>1==R</apll>) <apll>R</apll><br />
<apll>  R ←→ &gt;    &lt;   R</apll> &nbsp;&nbsp;for all <apll>R</apll> with <apll>(¯1↑⍴  R)∊1 2 4 8</apll><br />
<apll>1/R ←→ &gt;    &lt;    R</apll> &nbsp;&nbsp;for all <apll>R</apll> with <apll>(¯1↑⍴1/R)∊1 2 4 8</apll><br />
<apll>  R ←→ &gt;[X] &lt;[X] R</apll> &nbsp;&nbsp;for all non-scalar <apll>R</apll> with <apll>(⍴R)[X]∊1 2 4 8</apll><br />


== Acknowledgements==
== Acknowledgements==


<p>This symbol and its name were suggested by David A. Rabenhorst.</p>
<p>This symbol and its name were suggested by David A. Rabenhorst.</p>

Latest revision as of 16:14, 15 April 2018

Z←<R    or    Z←<[X] R converts R to a Hypercomplex array if (⍴R)[X]∊2 4 8 or a Real array if (⍴R)[X]=1.
X is an optional numeric singleton axis with X∊⍳⍴⍴R &mddash; otherwise (including if R is a scalar), an AXIS ERROR is signalled. If [X] is omitted, the operation applies to the last axis.
R is an arbitrary Real numeric array, i.e., Fixed-Precision Boolean, Integer, Floating Point, Arithmetic Progression Array, or Multiple-Precision Integer/Rational or Floating Point — otherwise, a DOMAIN ERROR is signalled. The X-axis length (⍴R)[X] must be 1, 2, 4, or 8 — otherwise, a LENGTH ERROR is signalled.
Z is the corresponding Real or Hypercomplex array of shape (X≠⍳⍴⍴R)/⍴R using the numbers along the X-axis of R as the coefficients of the resulting Real or Hypercomplex array. If
  • (⍴R)[X] is 1, the result is an array of Real numbers,
  • (⍴R)[X] is 2, the result is an array of Complex numbers,
  • (⍴R)[X] is 4, the result is an array of Quaternion numbers,
  • (⍴R)[X] is 8, the result is an array of Octonion numbers.


For example,

      <23 23       <10 20 10J20       <2 4⍴⍳8 1i2j3k4 5i6j7k8       <[1] 2 4⍴⍳8 1J5 2J6 3J7 4J8       <2 8⍴(⍳8),⌽⍳8 1i2j3k4l5ij6jk7kl8 8i7j6k5l4ij3jk2kl1       <[1] 2 8⍴(⍳8),⌽⍳8 1J8 2J7 3J6 4J5 5J4 6J3 7J2 8J1       ⍴⎕←<2 3 1⍴⍳6 1 2 3 4 5 6 2 3       ⍴⎕←<2 3 2⍴(⍳6),⌽⍳6 1J2 3J4 5J6 6J5 4J3 2J1 2 3       ⍴⎕←<[1] 2 3 2⍴(⍳6),⌽⍳6 1J6 2J5 3J4 4J3 5J2 6J1 3 2       ⍴⎕←<2 3 4⍴(⍳12),⌽⍳12  1i2j3k4   5i6j7k8 9i10j11k12 12i11j10k9 8i7j6k5 4i3j2k1 2 3       ⍴⎕←<2 3 8⍴(⍳24),⌽⍳24  1i2j3k4l5ij6jk7kl8         9i10j11k12l13ij14jk15kl16 17i18j19k20l21ij22jk23kl24 24i23j22k21l20ij19jk18kl17 16i15j14k13l12ij11jk10kl9   8i7j6k5l4ij3jk2kl1 2 3

Identities

R ←→ < > R   for all Real (1==R) R (see Dilate for the definition of monadic Right Caret)
R ←→ <[X] >[X] R   for all Real (1==R) R
R ←→ > < R   for all R with (¯1↑⍴ R)∊1 2 4 8
1/R ←→ > < R   for all R with (¯1↑⍴1/R)∊1 2 4 8
R ←→ >[X] <[X] R   for all non-scalar R with (⍴R)[X]∊1 2 4 8

Acknowledgements

This symbol and its name were suggested by David A. Rabenhorst.

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