Difference between revisions of "SingularValueDecomp"
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== SingularValueDecomp(a, i, j, j2) == | == 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 ''[[ | + | [[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 | ||
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: | 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: | ||
| − | :<code>Variable A := Sum(Sum(U*W, J)*Transpose(V, J, J2), J2)</code> | + | :<code>Variable A := [[Sum]]([[Sum]](U*W, J)*[[Transpose]](V, J, J2), J2)</code> |
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 <code>SvdIndex</code> with each element, ''U'', ''W'', and ''V'', being a reference to the corresponding array. | 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 <code>SvdIndex</code> with each element, ''U'', ''W'', and ''V'', being a reference to the corresponding array. | ||
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Use the [[Using References|# (dereference) operator]] to obtain the matrix value from each reference, as in: | Use the [[Using References|# (dereference) operator]] to obtain the matrix value from each reference, as in: | ||
| − | :<code>Index J2 := CopyIndex(J)</code> | + | :<code>Index J2 := [[CopyIndex]](J)</code> |
| − | :<code>Variable SvdResult := SingularValueDecomp(A, I, J, J2)</code> | + | :<code>Variable SvdResult := [[SingularValueDecomp]](A, I, J, J2)</code> |
:<code>Variable U := #SvdResult[SvdIndex = 'U']</code> | :<code>Variable U := #SvdResult[SvdIndex = 'U']</code> | ||
:<code>Variable W := #SvdResult[SvdIndex = 'W']</code> | :<code>Variable W := #SvdResult[SvdIndex = 'W']</code> | ||
:<code>Variable V := #SvdResult[SvdIndex = 'V']</code> | :<code>Variable V := #SvdResult[SvdIndex = 'V']</code> | ||
| + | |||
| + | == Matrix inverse == | ||
| + | The inverse of a square matrix A, in Analytica syntax, is | ||
| + | <code> | ||
| + | :[[Var]] Winv := [[If]] J=J2 [[Then]] 1/W [[Else]] W; | ||
| + | :[[Transpose]]([[Sum]]([[Sum]](U*Winv, J)*[[Transpose]](V, J, J2), J2),I,J) | ||
| + | </code> | ||
| + | |||
| + | 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 <code>W</code> will be very close to zero. To avoid the numerical instabilities, the diagonal entries corresponding to the very small <code>W</code> can be replaced with 0 in <code>Winv</code>: | ||
| + | |||
| + | <code> | ||
| + | :[[Var]] 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> | ||
| + | |||
== See Also == | == See Also == | ||
Revision as of 15:42, 14 May 2018
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
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:
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
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:
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