Symbolic derivatives with tomSym - Revision history

Per at 12:15, 28 July 2011

2011-07-28T12:15:18Z

← Older revision		Revision as of 07:15, 28 July 2011
Line 1:		Line 1:
	The main strength of [[tomSym]] is its ability to automatically and quickly compute symbolic derivatives of matrix expressions. ~~The~~ derivatives can then be converted into efficient Matlab code.		The main strength of [[tomSym]] is its ability to automatically and quickly compute symbolic derivatives of matrix expressions. These derivatives can then be converted into efficient Matlab code.

	== Notation ==		== Notation ==
Line 67:		Line 67:
	tempC2 = 2*y;		tempC2 = 2*y;
	out = scalerows(tempC2(Ax).^(tempC2-1),kron(x',speye(2)));		out = scalerows(tempC2(Ax).^(tempC2-1),kron(x',speye(2)));

			== Alternatives ==

			There are two other forms of derivatives available in TOMLAB: Automatic Derivatives using [[MAD]], and finite differences. It is also possible to mix the techniques by using tomSym as the top layer, and using MAD or finite differences to compute the derivatives of a subexpression using the <code>wrap</code> function.

			The main advantage of tomSym over MAD and finite differeces is the execution speed once the code has been generated, but tomSym cannot handle all kinds of functions efficiently. With functions containing large for-loops, it is often better to use one of the other alternatives.

	[[Category:tomSym]]		[[Category:tomSym]]

Per at 07:33, 28 July 2011

2011-07-28T07:33:49Z

← Older revision		Revision as of 02:33, 28 July 2011
Line 67:		Line 67:
	tempC2 = 2*y;		tempC2 = 2*y;
	out = scalerows(tempC2(Ax).^(tempC2-1),kron(x',speye(2)));		out = scalerows(tempC2(Ax).^(tempC2-1),kron(x',speye(2)));

			[[Category:tomSym]]

Per at 07:27, 28 July 2011

2011-07-28T07:27:15Z

← Older revision		Revision as of 02:27, 28 July 2011
Line 1:		Line 1:
	The main strength of tomSym is its ability to automatically and quickly compute symbolic derivatives of matrix expressions. The derivatives can then be converted into efficient Matlab code.		The main strength of [[tomSym]] is its ability to automatically and quickly compute symbolic derivatives of matrix expressions. The derivatives can then be converted into efficient Matlab code.

	== Notation ==		== Notation ==

Per: /* Example */

2011-07-28T07:14:20Z

Example

← Older revision		Revision as of 02:14, 28 July 2011
Line 60:		Line 60:
	the identity matrix is generated using <code>speye</code> in m-code, but displayed as <code>eye</code>.		the identity matrix is generated using <code>speye</code> in m-code, but displayed as <code>eye</code>.
	(Derivatives tend to involve many sparse matrices, which Matlab handles efficiently.)		(Derivatives tend to involve many sparse matrices, which Matlab handles efficiently.)
	Also, subexpressions that occur ~~more than once~~ are broken out and computed only once.		Also, subexpressions that occur multiple times are broken out and computed only once.
	The <code>mcode</code> command converts a tomSym object to a matlab code string.		The <code>mcode</code> command converts a tomSym object to a matlab code string.

Per: /* Example */

2011-07-28T07:13:56Z

Example

← Older revision		Revision as of 02:13, 28 July 2011
Line 60:		Line 60:
	the identity matrix is generated using <code>speye</code> in m-code, but displayed as <code>eye</code>.		the identity matrix is generated using <code>speye</code> in m-code, but displayed as <code>eye</code>.
	(Derivatives tend to involve many sparse matrices, which Matlab handles efficiently.)		(Derivatives tend to involve many sparse matrices, which Matlab handles efficiently.)
	Also, subexpressions that occur more than once are broken out and computed ~~separately~~.		Also, subexpressions that occur more than once are broken out and computed only once.
	The <code>mcode</code> command converts a tomSym object to a matlab code string.		The <code>mcode</code> command converts a tomSym object to a matlab code string.

Per at 07:11, 28 July 2011

2011-07-28T07:11:31Z

← Older revision		Revision as of 02:11, 28 July 2011
Line 42:		Line 42:
	Derivatives are represented using matrix operations on matrix symbols. This yields much faster code compared		Derivatives are represented using matrix operations on matrix symbols. This yields much faster code compared
	to other symbolic software which represents each element of a matrix with its own symbol. TomSym is routinely		to other symbolic software which represents each element of a matrix with its own symbol. TomSym is routinely
	used to solve optimization problems involving thousands of variables, and ~~symbolic~~ matrices with millions of elements.		used to solve optimization problems involving thousands of variables, and matrices with millions of elements.

	== Example ==		== Example ==
Line 57:		Line 57:
	In the above example, the 2x1 symbol ''f'' is differentiated w.r.t the 2x3 symbol ''A''. The result is a 2x6 matrix, representing d(vec(''f''))/d(vec(''A'')).		In the above example, the 2x1 symbol ''f'' is differentiated w.r.t the 2x3 symbol ''A''. The result is a 2x6 matrix, representing d(vec(''f''))/d(vec(''A'')).

	The displayed text is not necessarily identical to the m-code that will be generated from an expression. For example, subexpressions that occur more than once are broken out and computed separately. The <code>mcode</code> command converts a tomSym object to a matlab code string.		The displayed text is not necessarily identical to the m-code that will be generated from an expression. For example,
			the identity matrix is generated using <code>speye</code> in m-code, but displayed as <code>eye</code>.
			(Derivatives tend to involve many sparse matrices, which Matlab handles efficiently.)
			Also, subexpressions that occur more than once are broken out and computed separately.
			The <code>mcode</code> command converts a tomSym object to a matlab code string.

	>> mcode(g)		>> mcode(g)
	ans =		ans =
	tempC2 = 2*y;		tempC2 = 2*y;
	out = scalerows(tempC2(Ax).^(tempC2-1),kron(x',~~tempD{1}~~));		out = scalerows(tempC2(Ax).^(tempC2-1),kron(x',speye(2)));

Per: /* How derivatives are generated */

2011-07-28T06:59:27Z

How derivatives are generated

← Older revision		Revision as of 01:59, 28 July 2011
Line 42:		Line 42:
	Derivatives are represented using matrix operations on matrix symbols. This yields much faster code compared		Derivatives are represented using matrix operations on matrix symbols. This yields much faster code compared
	to other symbolic software which represents each element of a matrix with its own symbol. TomSym is routinely		to other symbolic software which represents each element of a matrix with its own symbol. TomSym is routinely
	used to solve optimization problems involving thousands of variables, and matrices with millions of elements.		used to solve optimization problems involving thousands of variables, and symbolic matrices with millions of elements.

	== Example ==		== Example ==

Per: /* Example */

2011-07-28T06:55:55Z

Example

← Older revision		Revision as of 01:55, 28 July 2011
Line 49:		Line 49:
	>> toms 2x3 A		>> toms 2x3 A
	>> f = (Ax).^(2y)		>> f = (Ax).^(2y)

	f = tomSym(2x1):		f = tomSym(2x1):
			(Ax).^(2y)
	(Ax).^(2y)		>> g = derivative(f,A)
			g = tomSym(2x6):
	>> derivative(f,A)		scalerows((2y)(Ax).^(2y-1),kron(x',eye(2)))

	~~ans~~ = tomSym(2x6):

	(2y)~~setdiag(~~(Ax).^(2y-1))*kron(x',eye(2))

	In the above example, the 2x1 symbol ''f'' is differentiated w.r.t the 2x3 symbol ''A''. The result is a 2x6 matrix, representing d(vec(''f''))/d(vec(''A'')).		In the above example, the 2x1 symbol ''f'' is differentiated w.r.t the 2x3 symbol ''A''. The result is a 2x6 matrix, representing d(vec(''f''))/d(vec(''A'')).

	The displayed text is not necessarily identical to the m-code that will be generated from an expression. For example, ~~the identity matrix is generated using <code>speye</code> in m-code, but displayed as <code>eye</code>. (Derivatives tend to involve many sparse matrices, which Matlab handles efficiently~~.) The <code>~~mcodestr~~</code> command converts a tomSym object to a matlab code string.		The displayed text is not necessarily identical to the m-code that will be generated from an expression. For example, subexpressions that occur more than once are broken out and computed separately. The <code>mcode</code> command converts a tomSym object to a matlab code string.

	>> ~~mcodestr~~(~~ans~~)		>> mcode(g)
	ans =		ans =
	(2y)~~setdiag(~~(Ax).^(~~2y~~-1))*kron(x',~~speye(2~~))		tempC2 = 2*y;
			out = scalerows(tempC2(Ax).^(tempC2-1),kron(x',tempD{1}));

Per: /* How derivatives are generated */

2011-07-28T06:50:41Z

How derivatives are generated

← Older revision		Revision as of 01:50, 28 July 2011
Line 34:		Line 34:
	TomSym's <tt>derivative</tt> and <tt>derivatives</tt> functions generates symbolic Jacobians for each subexpression,		TomSym's <tt>derivative</tt> and <tt>derivatives</tt> functions generates symbolic Jacobians for each subexpression,
	and then multiplies these together using the chain rule. (In tomSym notation, <tt>derivative(A,A)</tt> is the identity		and then multiplies these together using the chain rule. (In tomSym notation, <tt>derivative(A,A)</tt> is the identity
	matrix, so the chain rule applies using normal matrix multiplication.)		matrix, so the chain rule applies using normal matrix multiplication.)
			Some simplifications are performed along the way, in order to reduce memory requirements and improve speed.

	The accumulation can be done in either "forwards" or "backwards" mode.		The accumulation can be done in either "forwards" or "backwards" mode.

Per: How derivatives are generated

2011-07-28T06:47:37Z

How derivatives are generated

@@ Line 30: / Line 30: @@
 The difference in notation only involves the ordering of the elements, and reordering the elements to a different notational convention should be trivial if tomSym is used to generate derivatives for applications other than TOMLAB optimization.
-=== Example ===
+== How derivatives are generated ==
   >> toms     y
   >> toms 3x1 x