class SVD4 extends SVDecomp with Error
The SVD4 class is used to compute the Singular Value Decomposition 'SVD' of
matrix 'aa' using the Golub-Kahan-Reinsch Algorithm.
Factor/decompose matrix 'aa' into the product of three matrices:
aa = uu * a * vv.t
where 'uu' is a matrix of orthogonal eigenvectors of 'aa * aa.t' (LEFT SINGULAR VECTORS) 'vv' is a matrix of orthogonal eigenvectors of 'aa.t * aa' (RIGHT SINGULAR VECTORS) and 'a' is a diagonal matrix of square roots of eigenvalues of 'aa.t * aa' & 'aa * aa.t' (SINGULAR VALUES). FIX: need to reorder so singular values are in decreasing order. FIX: make the singular values positive ------------------------------------------------------------------------------
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Instance Constructors
Type Members
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type
FactorType = (MatriD, VectoD, MatriD)
Factor type contains 'u, s, v' which are the left orthogonal matrix, the diagonal matrix/vector containing singular values and the right orthogonal matrix.
Factor type contains 'u, s, v' which are the left orthogonal matrix, the diagonal matrix/vector containing singular values and the right orthogonal matrix.
- Definition Classes
- SVDecomp
-
type
FactorTypeFull = (MatriD, MatriD, MatriD)
- Definition Classes
- SVDecomp
Value Members
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def
conditionNum: Double
Compute the condition number of 'this' matrix, i.e., the ratio of the largest singular value to the smallest.
Compute the condition number of 'this' matrix, i.e., the ratio of the largest singular value to the smallest. Note, if not of full rank, it will be infinity.
- Definition Classes
- SVDecomp
-
def
factor(): SVDecomp
Factor/deflate the matrix by iteratively turning elements not in the main diagonal to zero.
Factor/deflate the matrix by iteratively turning elements not in the main diagonal to zero.
- Definition Classes
- SVDecomp → Factorization
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def
factor1(): MatriD
Factor a matrix into the product of two matrices, e.g., 'a = l * l.t', returning only the first matrix.
Factor a matrix into the product of two matrices, e.g., 'a = l * l.t', returning only the first matrix.
- Definition Classes
- Factorization
-
def
factor12(): (MatriD, MatriD)
Factor a matrix into the product of two matrices, e.g., 'a = l * l.t' or a = q * r, returning both the first and second matrices.
Factor a matrix into the product of two matrices, e.g., 'a = l * l.t' or a = q * r, returning both the first and second matrices.
- Definition Classes
- Factorization
-
def
factor123(): FactorType
Factor matrix 'a' into the product of a matrix of left singular vectors 'uu', a vector of singular values 's' and a matrix of right singular vectors 'vv' such that 'a = uu ** s * vv.t'.
-
def
factor2(): MatriD
Factor a matrix into the product of two matrices, e.g., 'a = l * l.t', returning only the second matrix.
Factor a matrix into the product of two matrices, e.g., 'a = l * l.t', returning only the second matrix.
- Definition Classes
- Factorization
-
def
factors: (MatriD, MatriD)
Return the two factored matrices.
Return the two factored matrices.
- Definition Classes
- SVDecomp → Factorization
-
final
def
flaw(method: String, message: String): Unit
Show the flaw by printing the error message.
Show the flaw by printing the error message.
- method
the method where the error occurred
- message
the error message
- Definition Classes
- Error
-
def
flip(u: MatriD, v: MatriD): Unit
Flip negative main diagonal elements in the singular vectors to positive.
Flip negative main diagonal elements in the singular vectors to positive.
- u
the left orthongonal matrix
- v
the right orthongonal matrix
- Definition Classes
- SVDecomp
-
def
flip(u: MatriD, s: VectoD): Unit
Flip negative singular values to positive and set singular values close to zero to zero.
Flip negative singular values to positive and set singular values close to zero to zero.
- u
the left orthongonal matrix
- s
the vector of singular values
- Definition Classes
- SVDecomp
-
def
reorder(ft: FactorType): Unit
Reorder the singular values to be in non-increasing order.
Reorder the singular values to be in non-increasing order. Must swap singular vectors in lock step with singular values. To minimize the number of swaps, selection sort is used.
- ft
the factored matrix (u, s, v)
- Definition Classes
- SVDecomp
-
def
solve(b: VectorD): VectoD
Solve for
xina^t*a*x = busingSVD.Solve for
xina^t*a*x = busingSVD.- b
the constant vector
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def
solve(b: VectoD): VectoD
Solve for 'x' in 'a^t*a*x = b' using
SVD.Solve for 'x' in 'a^t*a*x = b' using
SVD.- b
the constant vector
- Definition Classes
- SVDecomp → Factorization