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doc ///
Key
QRDecomposition
(QRDecomposition, MutableMatrix)
(QRDecomposition, Matrix)
Headline
compute a QR decomposition of a real matrix
Usage
(Q,R) = QRDecomposition A
Inputs
A:Matrix
or @ofClass MutableMatrix@, over $RR_{53}$
Outputs
Q:Matrix
R:Matrix
$(Q,R)$ is the QR-decomposition of $A$
Description
Text
If $A$ is a $m \times n$ matrix whose
columns are linearly independent, the $QR$ decomposition of $A$
is $QR = A$, where $Q$ is an $m \times m$ orthogonal matrix and
$R$ is an upper triangular $m \times n$ matrix.
Example
A = random(RR^5, RR^3)
(Q,R) = QRDecomposition A
Text
$R$ is upper triangular, and $Q$ is (close to) orthogonal.
Example
R
(transpose Q) * Q
clean(1e-10, oo)
R - (transpose Q) * A
clean(1e-10, oo)
Text
If the input is a @TO "MutableMatrix"@, then so are the output matrices.
Example
A = mutableMatrix(RR_53, 13, 5);
fillMatrix A
(Q,R) = QRDecomposition A
Q*R-A
clean(1e-10,oo)
Text
This function works by calling lapack routines, and so only uses the first 53 bits of precision.
Lapack also has a way of returning an encoded pair of matrices that contain
enough information to reconstruct $Q, R$.
Caveat
If the matrices are over higher precision real or complex fields, such as $RR_100$, this
extra precision is not used in the computation
SeeAlso
clean
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