class Fac_QR_H3 extends Fac_QR[MatrixD]
The Fac_QR_H3 class provides methods to factor an 'm-by-n' matrix 'a' into the
product of two matrices:
'q' - an 'm-by-n' orthogonal matrix and 'r' - an 'n-by-n' right upper triangular matrix
such that 'a = q * r'. It uses Householder orthogonalization. Note, orthogonal means that 'q.t * q = I'.
- See also
www.stat.wisc.edu/~larget/math496/qr.html ------------------------------------------------------------------------------ This implementation is the easiest to understandard, but the least efficient. Caveat: for m < n use
Fac_LQ. FIX: change 'aa: MatrixD' to 'aa: MatriD', requires 'times_ip_pre' in trait ------------------------------------------------------------------------------QRDecomposition.java in Jama
5.1 and 5.2 in Matrix Computations
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def
computeQ(): Unit
Compute the full 'q' orthogonal matrix based on updated values in 'a'.
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def
factor(): Fac_QR_H3
Factor matrix 'a' into the product of two matrices 'a = q * r', where 'q' is an orthogonal matrix and 'r' is a right upper triangular matrix.
Factor matrix 'a' into the product of two matrices 'a = q * r', where 'q' is an orthogonal matrix and 'r' is a right upper triangular matrix.
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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.
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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.
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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.
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val
factored: Boolean
Flag indicating whether the matrix has been factored
Flag indicating whether the matrix has been factored
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def
factors: (MatriD, MatriD)
Return both the orthogonal 'q' matrix and the right upper triangular 'r' matrix.
Return both the orthogonal 'q' matrix and the right upper triangular 'r' matrix.
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Show the flaw by printing the error message.
Show the flaw by printing the error message.
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the error message
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def
hReflect(v: VectoD): MatrixD
Return the Householder reflection matrix 'h' computed from Householder vector 'v'.
Return the Householder reflection matrix 'h' computed from Householder vector 'v'.
- v
the Householder vector
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def
houseV(x: VectoD, α: Double): VectoD
Return the Householder vector 'v' computed from vector 'x'.
Return the Householder vector 'v' computed from vector 'x'.
- x
the given vector for calculating the Householder vector
- α
the sign adjusted norm of x
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isInstanceOf[T0]: Boolean
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val
m: Int
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val
n: Int
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def
nullspace(rank: Int): MatriD
Compute the nullspace of matrix 'a': basis { x | a*x = 0 }' using 'QR' Factorization 'q*r*x = 0'.
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def
nullspaceV: VectorD
Compute the nullspace of matrix 'a: { x | a*x = 0 }' using 'QR' Factorization 'q*r*x = 0'.
Compute the nullspace of matrix 'a: { x | a*x = 0 }' using 'QR' Factorization 'q*r*x = 0'. Gives only one vector in the nullspace.
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val
p: Int
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val
q: MatrixD
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val
r: MatriD
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def
solve(b: VectoD): VectoD
Solve for 'x' in 'aa*x = b' using the QR Factorization 'aa = q*r' via 'r*x = q.t * b'.
Solve for 'x' in 'aa*x = b' using the QR Factorization 'aa = q*r' via 'r*x = q.t * b'. Requires calculating 'q' matrix first.
- b
the constant vector@param y the constant vector
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