laic1(3)

Library Functions Manual

laic1(3)

NAME

laic1 - laic1: condition estimate, step in gelsy

SYNOPSIS

Functions

subroutine claic1 (job, j, x, sest, w, gamma, sestpr, s, c)
CLAIC1 applies one step of incremental condition estimation. subroutine dlaic1 (job, j, x, sest, w, gamma, sestpr, s, c)
DLAIC1 applies one step of incremental condition estimation. subroutine slaic1 (job, j, x, sest, w, gamma, sestpr, s, c)
SLAIC1 applies one step of incremental condition estimation. subroutine zlaic1 (job, j, x, sest, w, gamma, sestpr, s, c)
ZLAIC1 applies one step of incremental condition estimation.

Detailed Description

Function Documentation

subroutine claic1 (integer job, integer j, complex, dimension( j ) x, real sest, complex, dimension( j ) w, complex gamma, real sestpr, complex s, complex c)

CLAIC1 applies one step of incremental condition estimation.

Purpose:

!>
!> 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**H 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]**H and sestpr**2 is an eigenpair of the system
!>
!>     diag(sest*sest, 0) + [alpha  gamma] * [ conjg(alpha) ]
!>                                           [ conjg(gamma) ]
!>
!> where  alpha =  x**H*w.
!>

Parameters

JOB

!>          JOB is INTEGER
!>          = 1: an estimate for the largest singular value is computed.
!>          = 2: an estimate for the smallest singular value is computed.
!>

!>          J is INTEGER
!>          Length of X and W
!>

!>          X is COMPLEX array, dimension (J)
!>          The j-vector x.
!>

SEST

!>          SEST is REAL
!>          Estimated singular value of j by j matrix L
!>

!>          W is COMPLEX array, dimension (J)
!>          The j-vector w.
!>

GAMMA

!>          GAMMA is COMPLEX
!>          The diagonal element gamma.
!>

SESTPR

!>          SESTPR is REAL
!>          Estimated singular value of (j+1) by (j+1) matrix Lhat.
!>

!>          S is COMPLEX
!>          Sine needed in forming xhat.
!>

!>          C is COMPLEX
!>          Cosine needed in forming xhat.
!>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 134 of file claic1.f.

subroutine dlaic1 (integer job, integer j, double precision, dimension( j ) x, double precision sest, double precision, dimension( j ) w, double precision gamma, double precision sestpr, double precision s, double precision c)

DLAIC1 applies one step of incremental condition estimation.

Purpose:

!>
!> DLAIC1 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 DLAIC1 computes sestpr, s, c such that
!> the vector
!>                 [ s*x ]
!>          xhat = [  c  ]
!> is an approximate singular vector of
!>                 [ L       0  ]
!>          Lhat = [ w**T 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]**T and sestpr**2 is an eigenpair of the system
!>
!>     diag(sest*sest, 0) + [alpha  gamma] * [ alpha ]
!>                                           [ gamma ]
!>
!> where  alpha =  x**T*w.
!>

Parameters

JOB

!>          JOB is INTEGER
!>          = 1: an estimate for the largest singular value is computed.
!>          = 2: an estimate for the smallest singular value is computed.
!>

!>          J is INTEGER
!>          Length of X and W
!>

!>          X is DOUBLE PRECISION array, dimension (J)
!>          The j-vector x.
!>

SEST

!>          SEST is DOUBLE PRECISION
!>          Estimated singular value of j by j matrix L
!>

!>          W is DOUBLE PRECISION array, dimension (J)
!>          The j-vector w.
!>

GAMMA

!>          GAMMA is DOUBLE PRECISION
!>          The diagonal element gamma.
!>

SESTPR

!>          SESTPR is DOUBLE PRECISION
!>          Estimated singular value of (j+1) by (j+1) matrix Lhat.
!>

!>          S is DOUBLE PRECISION
!>          Sine needed in forming xhat.
!>

!>          C is DOUBLE PRECISION
!>          Cosine needed in forming xhat.
!>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 133 of file dlaic1.f.

subroutine slaic1 (integer job, integer j, real, dimension( j ) x, real sest, real, dimension( j ) w, real gamma, real sestpr, real s, real c)

SLAIC1 applies one step of incremental condition estimation.

Purpose:

!>
!> SLAIC1 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 SLAIC1 computes sestpr, s, c such that
!> the vector
!>                 [ s*x ]
!>          xhat = [  c  ]
!> is an approximate singular vector of
!>                 [ L      0  ]
!>          Lhat = [ w**T 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]**T and sestpr**2 is an eigenpair of the system
!>
!>     diag(sest*sest, 0) + [alpha  gamma] * [ alpha ]
!>                                           [ gamma ]
!>
!> where  alpha =  x**T*w.
!>

Parameters

JOB

!>          JOB is INTEGER
!>          = 1: an estimate for the largest singular value is computed.
!>          = 2: an estimate for the smallest singular value is computed.
!>

!>          J is INTEGER
!>          Length of X and W
!>

!>          X is REAL array, dimension (J)
!>          The j-vector x.
!>

SEST

!>          SEST is REAL
!>          Estimated singular value of j by j matrix L
!>

!>          W is REAL array, dimension (J)
!>          The j-vector w.
!>

GAMMA

!>          GAMMA is REAL
!>          The diagonal element gamma.
!>

SESTPR

!>          SESTPR is REAL
!>          Estimated singular value of (j+1) by (j+1) matrix Lhat.
!>

!>          S is REAL
!>          Sine needed in forming xhat.
!>

!>          C is REAL
!>          Cosine needed in forming xhat.
!>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 133 of file slaic1.f.

subroutine zlaic1 (integer job, integer j, complex16, dimension( j ) x, double precision sest, complex16, dimension( j ) w, complex16 gamma, double precision sestpr, complex16 s, complex*16 c)

ZLAIC1 applies one step of incremental condition estimation.

Purpose:

!>
!> ZLAIC1 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 ZLAIC1 computes sestpr, s, c such that
!> the vector
!>                 [ s*x ]
!>          xhat = [  c  ]
!> is an approximate singular vector of
!>                 [ L       0  ]
!>          Lhat = [ w**H 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]**H and sestpr**2 is an eigenpair of the system
!>
!>     diag(sest*sest, 0) + [alpha  gamma] * [ conjg(alpha) ]
!>                                           [ conjg(gamma) ]
!>
!> where  alpha =  x**H * w.
!>

Parameters

JOB

!>          JOB is INTEGER
!>          = 1: an estimate for the largest singular value is computed.
!>          = 2: an estimate for the smallest singular value is computed.
!>

!>          J is INTEGER
!>          Length of X and W
!>

!>          X is COMPLEX*16 array, dimension (J)
!>          The j-vector x.
!>

SEST

!>          SEST is DOUBLE PRECISION
!>          Estimated singular value of j by j matrix L
!>

!>          W is COMPLEX*16 array, dimension (J)
!>          The j-vector w.
!>

GAMMA

!>          GAMMA is COMPLEX*16
!>          The diagonal element gamma.
!>

SESTPR

!>          SESTPR is DOUBLE PRECISION
!>          Estimated singular value of (j+1) by (j+1) matrix Lhat.
!>

!>          S is COMPLEX*16
!>          Sine needed in forming xhat.
!>

!>          C is COMPLEX*16
!>          Cosine needed in forming xhat.
!>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 134 of file zlaic1.f.

Author

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