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←<R</apll></td> | <td valign="top"><apll>Z←<R</apll> or <apll>Z←<[X] R</apll></td> | ||
<td></td> | <td></td> | ||
<td></td> | <td></td> | ||
<td>converts <apll>R</apll> to a Hypercomplex array if <apll>( | <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> | <td><apll>X</apll> is an optional numeric singleton axis with <apll>X∊⍳⍴⍴R</apll>. If <apll>X</apll> is omitted, it defaults to the last axis.</td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td><apll>Z</apll> is the corresponding Real or Hypercomplex array of shape <apll> | <td><apll>R</apll> is an arbitrary Real numeric array (BOOL, INT, FLT, APA, RAT, VFP — otherwise, <apll>DOMAIN ERROR</apll>) whose <apll>X</apll>-axis length <apll>(⍴R)[X]</apll> is <apll>1</apll>, <apll>2</apll>, <apll>4</apll>, or <apll>8</apll> — otherwise, <apll>LENGTH ERROR</apll>.</td> | ||
</tr> | |||
<tr> | |||
<td><apll>Z</apll> is the corresponding Real or Hypercomplex array of shape <apll>(X≠⍳⍴⍴R)/⍴R</apll> using the items along the <apll>X</apll>-axis of <apll>R</apll> as the coefficients of the resulting Real or Hypercomplex array. If <apll>(⍴R)[X]</apll> is <apll>1</apll>, the result is the Real array <apll>((X≠⍳⍴⍴R)/⍴R)⍴R</apll>, if <apll>(⍴R)[X]</apll> is <apll>2</apll>, the result is a Complex array, if <apll>(⍴R)[X]</apll> is <apll>4</apll>, the result is a Quaternion array, and if <apll>(⍴R)[X]</apll> is <apll>8</apll>, the result is an Octonion array.</td> | |||
</tr> | </tr> | ||
</table> | </table> | ||
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<2 8⍴(⍳8),⌽⍳8<br /> | <2 8⍴(⍳8),⌽⍳8<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<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<br /> | ||
<[1] 2 8⍴(⍳8),⌽⍳8<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<br /> | |||
⍴⎕←<2 3 1⍴⍳6<br /> | ⍴⎕←<2 3 1⍴⍳6<br /> | ||
1 2 3<br /> | 1 2 3<br /> | ||
| Line 50: | Line 55: | ||
==Identities== | ==Identities== | ||
<apll>R ←→ <>R</apll> for all <apll>R</apll> (see [[Dilate]] for the definition of monadic Right Caret)<br /> | <apll>R ←→ <[X] >[X] R</apll> for all <apll>R</apll> (see [[Dilate]] for the definition of monadic Right Caret)<br /> | ||
<apll>R ←→ ><R</apll> for all <apll>R</apll> with <apll>( | <apll>R ←→ >[X] <[X] R</apll> for all <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> | ||
Revision as of 00:14, 12 April 2018
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| X is an optional numeric singleton axis with X∊⍳⍴⍴R. If X is omitted, it defaults to the last axis. | ||||
| R is an arbitrary Real numeric array (BOOL, INT, FLT, APA, RAT, VFP — otherwise, DOMAIN ERROR) whose X-axis length (⍴R)[X] is 1, 2, 4, or 8 — otherwise, LENGTH ERROR. | ||||
| Z is the corresponding Real or Hypercomplex array of shape (X≠⍳⍴⍴R)/⍴R using the items along the X-axis of R as the coefficients of the resulting Real or Hypercomplex array. If (⍴R)[X] is 1, the result is the Real array ((X≠⍳⍴⍴R)/⍴R)⍴R, if (⍴R)[X] is 2, the result is a Complex array, if (⍴R)[X] is 4, the result is a Quaternion array, and if (⍴R)[X] is 8, the result is an Octonion array. |
For example,
<23
23
<10 20
10J20
<2 4⍴⍳8
1i2j3k4 5i6j7k8
<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
⍴⎕←<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 ←→ <[X] >[X] R for all R (see Dilate for the definition of monadic Right Caret)
R ←→ >[X] <[X] R for all R with (⍴R)[X]∊1 2 4 8
Acknowledgements
This symbol and its name were suggested by David A. Rabenhorst.
