Ball Arithmetic - Revision history

WikiSysop: /* Contract */

2022-08-19T08:03:23Z

Contract

← Older revision		Revision as of 04:03, 19 August 2022
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	≤1.3 1<_E/>¯20		≤1.3 1<_E/>¯20
	1.3<_pom/>1<_E/>¯20		1.3<_pom/>1<_E/>¯20
	~~≤1r2~~ 1<_E/>¯20		≤1<_r/>2 1<_E/>¯20
	0.5<_pom/>1<_E/>¯20		0.5<_pom/>1<_E/>¯20
	≤○1		≤○1

WikiSysop: /* Mixed Notation */

2022-08-19T07:53:16Z

Mixed Notation

← Older revision		Revision as of 03:53, 19 August 2022
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	<apll><pre>		<apll><pre>
	1<_p/>1<_pom/>		1<_pi/>1<_pom/>
	3.141592653589793<_pom/>4.4<_E/>¯16		3.141592653589793<_pom/>4.4<_E/>¯16
	1<_x/>1<_pom/>		1<_x/>1<_pom/>

WikiSysop: /* Union and Intersection of Balls */

2022-08-19T07:46:35Z

Union and Intersection of Balls

← Older revision		Revision as of 03:46, 19 August 2022
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	<apll><pre>		<apll><pre>
	x←0.5<~~pom~~/>0.5 ⍝ The interval [0,1]		x←0.5<_pom/>0.5 ⍝ The interval [0,1]
	x∪≈x+2 ⍝ The interval [0,1] ∪ [2,3] ←→ [0,3]		x∪≈x+2 ⍝ The interval [0,1] ∪ [2,3] ←→ [0,3]
	1.5<~~pom~~/>1.5		1.5<_pom/>1.5
	x∪≈x+1 ⍝ The interval [0,1] ∪ [1,2] ←→ [0,2]		x∪≈x+1 ⍝ The interval [0,1] ∪ [1,2] ←→ [0,2]
	1<~~pom~~/>1		1<_pom/>1
	x∩≈x+1 ⍝ The interval [0,1] ∩ [1,2] ←→ [1,1]		x∩≈x+1 ⍝ The interval [0,1] ∩ [1,2] ←→ [1,1]
	1		1
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	</pre></apll>		</pre></apll>

	The identity element for <apll>∩≈</apll> is <apll>~~0±∞~~</apll>.		The identity element for <apll>∩≈</apll> is <apll>0<_pom/>∞</apll>.

	The derived functions are scalar.		The derived functions are scalar.

WikiSysop: /* Positional Relationships */

2019-03-19T14:57:11Z

Positional Relationships

← Older revision		Revision as of 10:57, 19 March 2019
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	Note the table is organized such that apart from the first row (whose '''Endpoint Relationship''' is Symmetric in <apll>L</apll> and <apll>R</apll>) the rows occur in six pairs where the '''Endpoint Relationship''' for each row in the pair can be obtained from the other by swapping <apll>L</apll> and <apll>R</apll>. The pairs of rows are ordered roughly by the amount of overlap between the two Balls.		Note the table is organized such that apart from the first row (whose '''Endpoint Relationship''' is Symmetric in <apll>L</apll> and <apll>R</apll>) the rows occur in six pairs where the '''Endpoint Relationship''' for each row in the pair can be obtained from the other by swapping <apll>L</apll> and <apll>R</apll>. The pairs of rows are ordered roughly by the amount of overlap between the two Balls.

	As you can see, we are missing '''Defining Expressions''' for several of the '''Relative Positions'''. I was hoping that this table would be filled in with more detail so it would serve as a measure of whether or not we had a complete set of functions/operators which could be used to tell apart all of the '''Relative Positions''' without having to resort to the '''Distinct Ball''' function <apll>{⍺∘≈⍵}</apll> constructed explicitly for this purpose. Please add to this table if you find a missing expression, or wish to propose whole new functions and/or ~~operators~~.		As you can see, we are missing '''Defining Expressions''' for several of the '''Relative Positions'''. I was hoping that this table would be filled in with more detail so it would serve as a measure of whether or not we had a complete set of functions/operators/hyperators which could be used to tell apart all of the '''Relative Positions''' without having to resort to the '''Distinct Ball''' function <apll>{⍺∘≈⍵}</apll> constructed explicitly for this purpose. Please add to this table if you find a missing expression, or wish to propose whole new functions, operators, and/or hyperators.

	If you would like to try a '''Defining Expression''' on the '''Sample Data''', collect the data into a <apll>13</apll> by <apll>2</apll> matrix, put your '''Defining Expression''' into (say) an Anonymous Function and reduce the same data matrix along the last coordinate — if you have a successful function, you should see a <apll>13</apll>-element Boolean vector with a single one in it. For example,		If you would like to try a '''Defining Expression''' on the '''Sample Data''', collect the data into a <apll>13</apll> by <apll>2</apll> matrix, put your '''Defining Expression''' into (say) an Anonymous Function and reduce the same data matrix along the last coordinate — if you have a successful function, you should see a <apll>13</apll>-element Boolean vector with a single one in it. For example,

WikiSysop: /* Conversion Between Balls and Intervals */

2019-01-10T22:18:02Z

Conversion Between Balls and Intervals

← Older revision		Revision as of 18:18, 10 January 2019
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	====Conversion Between Balls and Intervals====		====Conversion Between Balls and Intervals====

	~~At the moment, the monadic derived function is defined for two primitive functions.~~

	Because Ball Arithmetic has a close relationship with Interval Arithmetic, it may be desirable to convert between the two forms: Endpoint to Midpoint-Radius and its inverse. The resulting derived function is not scalar.		Because Ball Arithmetic has a close relationship with Interval Arithmetic, it may be desirable to convert between the two forms: Endpoint to Midpoint-Radius and its inverse. The resulting derived function is not scalar.

WikiSysop: /* Workspace Version Number */

2019-01-10T22:15:28Z

Workspace Version Number

← Older revision		Revision as of 18:15, 10 January 2019
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	==Workspace Version Number==		==Workspace Version Number==

	For the moment, if you save a workspace which contains a Ball Arithmetic number in a variable, the workspace version number is <apll>0.03</apll>. Otherwise, the version number is <apll>0.02</apll>. Workspaces with Ball Arithmetic Numbers ~~may not~~ be loaded by an earlier version of the interpreter.		For the moment, if you save a workspace which contains a Ball Arithmetic number in a variable, the workspace version number is <apll>0.03</apll>. Otherwise, the version number is <apll>0.02</apll>. Workspaces with Ball Arithmetic Numbers cannot be loaded by an earlier version of the interpreter.

	==Pending Design Decisions==		==Pending Design Decisions==

WikiSysop: /* Ball Arithmetic Numbers */

2019-01-10T22:13:26Z

Ball Arithmetic Numbers

← Older revision		Revision as of 18:13, 10 January 2019
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	A Ball Arithmetic number consist of two parts: a Midpoint and Radius. The Midpoint value represents the middle of the Ball and the Radius represents the distance around the Ball of possible values. There is no way to tell which value in the Ball is the correct answer — you should assume that all values are potential candidates.		A Ball Arithmetic number consist of two parts: a Midpoint and Radius. The Midpoint value represents the middle of the Ball and the Radius represents the distance around the Ball of possible values. There is no way to tell which value in the Ball is the correct answer — you should assume that all values are potential candidates.

	The precision at which the Midpoint ~~and Radius are~~ stored depends upon the current value of <apll>⎕FPC</apll>. For example, the following display		The precision at which the Midpoint is stored depends upon the current value of <apll>⎕FPC</apll> — the precision of the Radius is fixed at <apll>30</apll> bits. For example, the following display

	<apll><pre>		<apll><pre>
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	</pre></apll>		</pre></apll>

	indicates that the number <apll>÷3<_pom/></apll> has a Midpoint of <apll>0.3333333333333333</apll> at <apll>53</apll>-bit precision and a Radius of <apll>5.6<_E/>¯17</apll>. The Midpoint is stored as ~~a variable~~ precision number~~, and~~ the Radius is stored as a ~~low~~-precision number.		indicates that the number <apll>÷3<_pom/></apll> has a Midpoint of <apll>0.3333333333333333</apll> at <apll>53</apll>-bit precision and a Radius of <apll>5.6<_E/>¯17</apll>. The Midpoint is stored as an arbitrary precision number; the mantissa of the Radius is stored as a fixed-precision (<apll>30</apll> bit) number along with an exponent that may be arbitrarily large or small.

	Alternatively,		Alternatively,
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	</p>		</p>

	<p>		<p>The value of <apll>⎕FPC</apll> should be calculated so as to be sufficient to represent the Midpoint of all ball numbers being used. Otherwise, some results might be misleading.
	The ~~current~~ <apll>⎕FPC</apll> ~~precision~~
	should be sufficient to ~~cover both the midpoint and~~ the ~~radius~~
	of all ball numbers being used.
	Otherwise,
	some results might be misleading.
	</p>		</p>

	<p>		<p>For example,</p>
	For example,
	</p>
	<p><apll>		<p><apll>
	2<_v/>*~~⎕fpc←128~~		2<_v/>*⎕FPC←128
	3.402823669<_E/>38		3.402823669<_E/>38
	⍟ ⎕← * 1<_pom/>1<_e/>¯1000		⍟ ⎕← * 1<_pom/>1<_e/>¯1000
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	<p>		<p>
	What happened? -- <apll>⎕FPC</apll> was too small!		What happened — the logarithm of an exponential should return the original argument? In this case, it didn't because <apll>⎕FPC</apll> was too small!
	</p>		</p>

	<p><apll>		<p><apll>
	2<_v/>*~~⎕fpc←4096~~		2<_v/>*⎕FPC←4096
	1.044388881<_E/>1233		1.044388881<_E/>1233
	⍟ ⎕← * 1<_pom/>1<_e/>¯1000		⍟ ⎕← * 1<_pom/>1<_e/>¯1000
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	<p> </p>		<p> </p>

	A Ball describes a range of '''~~variable~~-precision floating point numbers''', and so the expression <apll>23<_pom/>1</apll> represents the set of Real numbers between <apll>22</apll> and <apll>24</apll>, inclusive, '''not''' the three integers <apll>22</apll>, <apll>23</apll>, <apll>24</apll>.		A Ball describes a range of '''arbitrary-precision floating point numbers''', and so the expression <apll>23<_pom/>1</apll> represents the set of Real numbers between <apll>22</apll> and <apll>24</apll>, inclusive, '''not''' the three integers <apll>22</apll>, <apll>23</apll>, <apll>24</apll>.

	<p class="note" style=" background-color:lightcyan;">In general, suffix notation on Ball Arithmetic constants should be sufficient for '''almost all purposes'''. That is, the Radius is already correctly calculated for constants such as <apll>23<_pom/></apll> whose Radius is zero, and <apll>1.2<_pom/></apll> whose Radius at <apll>53</apll> bits of precision is on the order of <apll>2*¯53</apll> (because of the inexactness of the Midpoint <apll>1.2</apll>) which displays as <apll>1.2<_pom/>2.2<_E/>¯16</apll>. Although the input notation supports it, the need to override the default Radius should be '''quite rare'''.</p>		<p class="note" style=" background-color:lightcyan;">In general, suffix notation on Ball Arithmetic constants should be sufficient for '''almost all purposes'''. That is, the Radius is already correctly calculated for constants such as <apll>23<_pom/></apll> whose Radius is zero, and <apll>1.2<_pom/></apll> whose Radius at <apll>53</apll> bits of precision is on the order of <apll>2*¯53</apll> (because of the inexactness of the Midpoint <apll>1.2</apll>) which displays as <apll>1.2<_pom/>2.2<_E/>¯16</apll>. Although the input notation supports it, the need to override the default Radius should be '''quite rare'''.</p>

WikiSysop: /* Hypercomplex Ball Arithmetic */

2019-01-10T15:39:23Z

Hypercomplex Ball Arithmetic

← Older revision		Revision as of 11:39, 10 January 2019
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	==Hypercomplex Ball Arithmetic==		==Hypercomplex Ball Arithmetic==

	Ball Arithmetic also extends to all Hypercomplex numbers (Complex, Quaternion, and Octonion) such as <apll>1.2<_pom/><_i/>2.2</apll> which represents a Complex Ball Arithmetic number whose '''Real''' part is <apll>1.2<_pom/></apll> and whose '''Imaginary''' part is <apll>2.2<_pom/></apll>. Note that the display of that number (at <apll>53</apll> bits of precision)		Real Ball Arithmetic also extends to all Hypercomplex numbers (Complex, Quaternion, and Octonion) such as <apll>1.2<_pom/><_i/>2.2</apll> which represents a Complex Ball Arithmetic number whose '''Real''' part is <apll>1.2<_pom/></apll> and whose '''Imaginary''' part is <apll>2.2<_pom/></apll>. Note that the display of that number (at <apll>53</apll> bits of precision)

	<apll><pre>		<apll><pre>
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	</pre></apll>		</pre></apll>

	demonstrates the inexactness of both ~~floating point~~ coefficients.		demonstrates the inexactness of both Real and Imaginary coefficients.

	==Small v. Large Radii==		==Small v. Large Radii==

WikiSysop: /* Containing Functions */

2019-01-09T18:30:37Z

Containing Functions

← Older revision		Revision as of 14:30, 9 January 2019
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	<apll><pre>		<apll><pre>
	(L R)~~←6±4 6±2~~		(L R)←6<_pom/>4 6<_pom/>2
	L⊃≈R		L⊃≈R
	1		1

WikiSysop: /* Conversion Between Balls and Intervals */

2019-01-09T18:29:16Z

Conversion Between Balls and Intervals

← Older revision		Revision as of 14:29, 9 January 2019
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	For example,		For example,

	<apll><pre> ≥≈ ~~2±0i0±1~~		<apll><pre> ≥≈ 2<_pom/>0<_i/>0<_pom/>1
	DOMAIN ERROR		DOMAIN ERROR
	≥≈ ~~2±0i0±1~~		≥≈ 2<_pom/>0<_i/>0<_pom/>1
	∧		∧
	≥≈>~~2±0i0±1~~		≥≈>2<_pom/>0<_i/>0<_pom/>1
	2 2		2 2
	¯1 1		¯1 1
	<≤≈≥≈>~~2±0i0±1~~		<≤≈≥≈>2<_pom/>0<_i/>0<_pom/>1
	~~2±0i0±1~~		2<_pom/>0<_i/>0<_pom/>1
	</pre></apll>		</pre></apll>