CLAIC1(3F) CLAIC1(3F)
CLAIC1 - applie one step of incremental condition estimation in its
simplest version
SUBROUTINE CLAIC1( JOB, J, X, SEST, W, GAMMA, SESTPR, S, C )
INTEGER J, JOB
REAL SEST, SESTPR
COMPLEX C, GAMMA, S
COMPLEX W( J ), X( J )
CLAIC1 applies one step of incremental condition estimation in its
simplest version:
Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j
lower triangular matrix L, such that
twonorm(L*x) = sest
Then CLAIC1 computes sestpr, s, c such that
the vector
[ s*x ]
xhat = [ c ]
is an approximate singular vector of
[ L 0 ]
Lhat = [ w' gamma ]
in the sense that
twonorm(Lhat*xhat) = sestpr.
Depending on JOB, an estimate for the largest or smallest singular value
is computed.
Note that [s c]' and sestpr**2 is an eigenpair of the system
diag(sest*sest, 0) + [alpha gamma] * [ conjg(alpha) ]
[ conjg(gamma) ]
where alpha = conjg(x)'*w.
JOB (input) INTEGER
= 1: an estimate for the largest singular value is computed.
= 2: an estimate for the smallest singular value is computed.
J (input) INTEGER
Length of X and W
Page 1
CLAIC1(3F) CLAIC1(3F)
X (input) COMPLEX array, dimension (J)
The j-vector x.
SEST (input) REAL
Estimated singular value of j by j matrix L
W (input) COMPLEX array, dimension (J)
The j-vector w.
GAMMA (input) COMPLEX
The diagonal element gamma.
SESTPR (output) REAL
Estimated singular value of (j+1) by (j+1) matrix Lhat.
S (output) COMPLEX
Sine needed in forming xhat.
C (output) COMPLEX
Cosine needed in forming xhat.
CLAIC1(3F) CLAIC1(3F)
CLAIC1 - applie one step of incremental condition estimation in its
simplest version
SUBROUTINE CLAIC1( JOB, J, X, SEST, W, GAMMA, SESTPR, S, C )
INTEGER J, JOB
REAL SEST, SESTPR
COMPLEX C, GAMMA, S
COMPLEX W( J ), X( J )
CLAIC1 applies one step of incremental condition estimation in its
simplest version:
Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j
lower triangular matrix L, such that
twonorm(L*x) = sest
Then CLAIC1 computes sestpr, s, c such that
the vector
[ s*x ]
xhat = [ c ]
is an approximate singular vector of
[ L 0 ]
Lhat = [ w' gamma ]
in the sense that
twonorm(Lhat*xhat) = sestpr.
Depending on JOB, an estimate for the largest or smallest singular value
is computed.
Note that [s c]' and sestpr**2 is an eigenpair of the system
diag(sest*sest, 0) + [alpha gamma] * [ conjg(alpha) ]
[ conjg(gamma) ]
where alpha = conjg(x)'*w.
JOB (input) INTEGER
= 1: an estimate for the largest singular value is computed.
= 2: an estimate for the smallest singular value is computed.
J (input) INTEGER
Length of X and W
Page 1
CLAIC1(3F) CLAIC1(3F)
X (input) COMPLEX array, dimension (J)
The j-vector x.
SEST (input) REAL
Estimated singular value of j by j matrix L
W (input) COMPLEX array, dimension (J)
The j-vector w.
GAMMA (input) COMPLEX
The diagonal element gamma.
SESTPR (output) REAL
Estimated singular value of (j+1) by (j+1) matrix Lhat.
S (output) COMPLEX
Sine needed in forming xhat.
C (output) COMPLEX
Cosine needed in forming xhat.
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