Параллельные вычисления в ИММ УрО РАН
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Next: Algorithmic Differences between LAPACK Up: Contents of ScaLAPACK Previous: Generalized Symmetric Definite Eigenproblems
Orthogonal or Unitary Matrices
A real orthogonal or complex unitary matrix (usually denoted Q) is often
represented in
ScaLAPACK as a product of elementary reflectors -- also referred to as
elementary Householder matrices (usually denoted ). For example,
The following details may occasionally be useful.
An elementary reflector (or elementary Householder matrix) H of order
n is a
unitary matrix of the form
Some redundancy in the representation (3.4) exists, which can be removed in various ways. Like LAPACK, the representation used in ScaLAPACK (which differs from that used in LINPACK or EISPACK) sets ; hence need not be stored. In real arithmetic, , except that implies H = I.
In complex arithmetic , may be
complex and satisfies
and .
Thus a complex H is
not Hermitian (as it is in other representations), but it is unitary,
which is the important property. The advantage of allowing to be
complex is that, given an arbitrary complex vector x, H can be computed
so that For further details, see Lehoucq [94].
Next: Algorithmic Differences between LAPACK Up: Contents of ScaLAPACK Previous: Generalized Symmetric Definite Eigenproblems Susan Blackford Tue May 13 09:21:01 EDT 1997 |