Difference between revisions of "SingularValueDecomp"

Revision as of 18:47, 24 May 2021

SingularValueDecomp(a, i, j, j2)

SingularValueDecomp computes the singular value decomposition of a matrix. Singular value decomposition is often used with sets of equations or matrices that are singular or ill-conditioned (that is, very close to singular). It factors a matrix «a», indexed by «i» and «j», with IndexLength(i) >= IndexLength(i), into three matrices, U, W, and V, such that:

a = U . W . V^T

where U and V are orthogonal matrices and W is a diagonal matrix. U is dimensioned by «i» and «j», W by «j» and «j2», and V by «j» and «j2». In Analytica notation:

Variable A := Sum(Sum(U*W, J)*Transpose(V, J, J2), J2)

The index «j2» must be the same size as «j» and is used to index the resulting W and V arrays. SingularValueDecomp returns an array of three elements indexed by a special system index named SvdIndex with each element, U, W, and V, being a reference to the corresponding array.

Use the # (dereference) operator to obtain the matrix value from each reference, as in:

Index J2 := CopyIndex(J)

Variable SvdResult := SingularValueDecomp(A, I, J, J2)

Variable U := #SvdResult[SvdIndex = 'U']

Variable W := #SvdResult[SvdIndex = 'W']

Variable V := #SvdResult[SvdIndex = 'V']

Matrix inverse

The inverse of a square matrix A, in Analytica syntax, is

Local Winv := If J=J2 Then 1/W Else 0;
Transpose(Sum(Sum(U*Winv, J)*Transpose(V, J, J2), J2),I,J)

Singular value decomposition can be used for matrix inverse when the matrix A is ill-conditioned, in which case the Invert function may encounter numeric instabilities. When the matrix is ill-conditioned (the Determinant is very close to zero), then some of the elements of the diagonal of W will be very close to zero. To avoid the numerical instabilities, the diagonal entries corresponding to the very small W can be replaced with 0 in Winv:

Local Winv := If J=J2 And Abs(W)>1e-4 Then 1/W Else 0;
Transpose(Sum(Sum(U*Winv, J)*Transpose(V, J, J2), J2),I,J)

To make this convenient to use, you can introduce a new User-Defined Function as follows:

Function MatInvert( A : [I,J] ; I,J : Index )

Definition:


Index J2 := J;
Local svd := SingularValueDecomp(A,I,J,J2);
Local U := #svd[SvdIndex='U'];
Local W := #svd[SvdIndex='W'];
Local Winv := if J=J2 And W>1e-5 then 1/W else 0;
Local V := #svd[SvdIndex='V'];
Transpose(Sum(Sum(U*Winv, J)*Transpose(V, J, J2), J2),I,J)

You can then use MatInvert(A,I,J) in place of Invert(A,I,J).

@@ Line 4: / Line 4: @@
 [[SingularValueDecomp]] computes the singular value decomposition of a matrix. Singular value decomposition is often used with sets of equations or matrices that are singular or ill-conditioned (that is, very close to singular). It factors a matrix «a», indexed by «i» and «j», with ''[[IndexLength]](i) >= [[IndexLength]](i)'', into three matrices, ''U'', ''W'', and ''V'', such that:
-:a = U . W . V
+:a = U . W . V<sup>T</sup>
 where ''U'' and ''V'' are orthogonal matrices and ''W'' is a diagonal matrix. ''U'' is dimensioned by «i» and «j», ''W'' by «j» and «j2», and ''V'' by «j» and «j2». In Analytica notation:
@@ Line 23: / Line 23: @@
 The inverse of a square matrix A, in Analytica syntax, is
 <code>
-:[[Var]] Winv := [[If]] J=J2 [[Then]] 1/W [[Else]] W;
+:[[Local]] Winv := [[If]] J=J2 [[Then]] 1/W [[Else]] 0;
 :[[Transpose]]([[Sum]]([[Sum]](U*Winv, J)*[[Transpose]](V, J, J2), J2),I,J)
 </code>
@@ Line 30: / Line 30: @@
 <code>
-:[[Var]] Winv := [[If]] J=J2 [[And]] [[Abs]](W)>1e-4 [[Then]] 1/W [[Else]] 0;
+:[[Local]] Winv := [[If]] J=J2 [[And]] [[Abs]](W)>1e-4 [[Then]] 1/W [[Else]] 0;
 :[[Transpose]]([[Sum]]([[Sum]](U*Winv, J)*[[Transpose]](V, J, J2), J2),I,J)
 </code>
+To make this convenient to use, you can introduce a new [[User-Defined Function]] as follows:
+:'''Function''' <code>MatInvert( A : [I,J] ; I,J : Index )</code>
+:'''Definition:'''<code>
+::Index J2 := J;
+::Local svd := SingularValueDecomp(A,I,J,J2);
+::Local U := #svd[SvdIndex='U'];
+::Local W := #svd[SvdIndex='W'];
+::Local Winv := if J=J2 And W>1e-5 then 1/W else 0;
+::Local V := #svd[SvdIndex='V'];
+::Transpose(Sum(Sum(U*Winv, J)*Transpose(V, J, J2), J2),I,J)
+</code>
+You can then use <code>MatInvert(A,I,J)</code> in place of <code>[[Invert]](A,I,J)</code>.
 == See Also ==