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linalg[QRdecomp] - QR decomposition of a matrix
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Calling Sequence
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QRdecomp(A)
QRdecomp(A, arg2, arg3, ...)
QRdecomp(A, Q='q', rank='r', fullspan=value)
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Parameters
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A
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rectangular matrix
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arg.i
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(optional) is of the form name=val
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rank='r'
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(optional) for returning the rank of A
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Q='q'
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(optional) for returning the Q factor of A
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fullspan=value
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(optional) include null span in Q
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Description
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The routine QRdecomp computes the QR decomposition of the matrix A.
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For matrices of floating-point entries, the numerically stable Householder-transformations are used. For symbolic computation, the Gram-Schmidt process is applied.
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The result is an upper triangular matrix R. The orthonormal (unitary) factor Q is passed back to the Q parameter.
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The default factorization is the full QR where R will have the same dimension as A. Q will be a full rank square matrix whose first n columns span the column space of A and whose last m-n columns span the null space of A.
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If the (optional) fullspan arg is set to false, a Q1R1 factorization will be given where the Q1 factor will have the same dimension as A and, assuming A has full column rank, the columns of Q will span the column space of A. The R factor will be square and agree in dimension with Q. The default for fullspan is true.
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If A is an n by n matrix then  .
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If A contains complex entries, the Q factor will be unitary.
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The command with(linalg,QRdecomp) allows the use of the abbreviated form of this command.
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Examples
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Important: The linalg package has been deprecated. Use the superseding packages, LinearAlgebra and VectorCalculus, instead.
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![A := matrix(3, 3, [1, 2, 3, 0, 0, 1, 2, 3, 4])](/support/helpjp/helpview.aspx?si=6236/file07280/math182.png)
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![A := matrix([[1, 2, 3], [0, 0, 1], [2, 3, 4]])](/support/helpjp/helpview.aspx?si=6236/file07280/math185.png)
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![R := matrix([[5^(1/2), (8/5)*5^(1/2), (11/5)*5^(1/2)], [0, (1/5)*5^(1/2), (2/5)*5^(1/2)], [0, 0, 1]])](/support/helpjp/helpview.aspx?si=6236/file07280/math192.png)
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Warning, unable to find a provably non-zero pivot
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![matrix([[(1/5)*5^(1/2), (2/5)*5^(1/2), 0], [0, 0, 1], [(2/5)*5^(1/2), -(1/5)*5^(1/2), 0]])](/support/helpjp/helpview.aspx?si=6236/file07280/math209.png)
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![A := matrix(4, 2, [1, 2, 3, 0, 0, 1, 2, 3])](/support/helpjp/helpview.aspx?si=6236/file07280/math213.png)
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![A := matrix([[1, 2], [3, 0], [0, 1], [2, 3]])](/support/helpjp/helpview.aspx?si=6236/file07280/math216.png)
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![R := matrix([[14^(1/2), (4/7)*14^(1/2)], [0, (1/7)*462^(1/2)]])](/support/helpjp/helpview.aspx?si=6236/file07280/math223.png)
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Warning, unable to find a provably non-zero pivot
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![matrix([[(1/14)*14^(1/2), (5/231)*462^(1/2)], [(3/14)*14^(1/2), -(2/77)*462^(1/2)], [0, (1/66)*462^(1/2)], [(1/7)*14^(1/2), (13/462)*462^(1/2)]])](/support/helpjp/helpview.aspx?si=6236/file07280/math240.png)
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