class Fac_QR_MGS extends Fac_QR[MatrixD]
The Fac_QR_MGS 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 Modified Gram-Schmidt (MGS) orthogonalization. Note, orthogonal means that 'q.t * q = I'.
- See also
http://en.wikipedia.org/wiki/Gram–Schmidt_process (stabilized Gram–Schmidt orthonormalization)
http://www.stat.wisc.edu/~larget/math496/qr.html
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def
computeQ(): Unit
Compute the full orthogonal matrix 'q'.
Compute the full orthogonal matrix 'q'. No implementation needed, since it is automatically computed.
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- Fac_QR_MGS → Fac_QR
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def
factor(): Fac_QR_MGS
Factor matrix 'a' into the product of two matrices, 'a = q * r', returning both the orthogonal 'q' matrix and the right upper triangular 'r' matrix.
Factor matrix 'a' into the product of two matrices, 'a = q * r', returning both the orthogonal 'q' matrix and the right upper triangular 'r' matrix. This algorithm uses Modified Gram-Schmidt 'MGS' orthogonalization.
- Definition Classes
- Fac_QR_MGS → Factorization
- See also
Algorithm 5.2.6 in Matrix Computations.
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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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- Factorization
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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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- Factorization
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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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- Factorization
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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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- Factorization
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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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- Fac_QR → Factorization
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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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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: { 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 a basis of dimension 'n - rank' for the nullspace
- rank
the rank of the matrix (number of linearly independent column vectors)
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- Fac_QR_MGS → Fac_QR
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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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- Fac_QR
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val
q: MatrixD
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- Fac_QR_MGS → Fac_QR
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val
r: MatriD
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- Fac_QR
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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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- Fac_QR → Factorization
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