SingularValueDecomp

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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 . VT

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).

See Also

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