.TH "lasd4" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME lasd4 \- lasd4: D&C step: secular equation nonlinear solver .SH SYNOPSIS .br .PP .SS "Functions" .in +1c .ti -1c .RI "subroutine \fBdlasd4\fP (n, i, d, z, delta, rho, sigma, work, info)" .br .RI "\fBDLASD4\fP computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix\&. Used by dbdsdc\&. " .ti -1c .RI "subroutine \fBslasd4\fP (n, i, d, z, delta, rho, sigma, work, info)" .br .RI "\fBSLASD4\fP computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix\&. Used by sbdsdc\&. " .in -1c .SH "Detailed Description" .PP .SH "Function Documentation" .PP .SS "subroutine dlasd4 (integer n, integer i, double precision, dimension( * ) d, double precision, dimension( * ) z, double precision, dimension( * ) delta, double precision rho, double precision sigma, double precision, dimension( * ) work, integer info)" .PP \fBDLASD4\fP computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix\&. Used by dbdsdc\&. .PP \fBPurpose:\fP .RS 4 .PP .nf This subroutine computes the square root of the I-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix whose entries are given as the squares of the corresponding entries in the array d, and that 0 <= D(i) < D(j) for i < j and that RHO > 0\&. This is arranged by the calling routine, and is no loss in generality\&. The rank-one modified system is thus diag( D ) * diag( D ) + RHO * Z * Z_transpose\&. where we assume the Euclidean norm of Z is 1\&. The method consists of approximating the rational functions in the secular equation by simpler interpolating rational functions\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIN\fP .PP .nf N is INTEGER The length of all arrays\&. .fi .PP .br \fII\fP .PP .nf I is INTEGER The index of the eigenvalue to be computed\&. 1 <= I <= N\&. .fi .PP .br \fID\fP .PP .nf D is DOUBLE PRECISION array, dimension ( N ) The original eigenvalues\&. It is assumed that they are in order, 0 <= D(I) < D(J) for I < J\&. .fi .PP .br \fIZ\fP .PP .nf Z is DOUBLE PRECISION array, dimension ( N ) The components of the updating vector\&. .fi .PP .br \fIDELTA\fP .PP .nf DELTA is DOUBLE PRECISION array, dimension ( N ) If N \&.ne\&. 1, DELTA contains (D(j) - sigma_I) in its j-th component\&. If N = 1, then DELTA(1) = 1\&. The vector DELTA contains the information necessary to construct the (singular) eigenvectors\&. .fi .PP .br \fIRHO\fP .PP .nf RHO is DOUBLE PRECISION The scalar in the symmetric updating formula\&. .fi .PP .br \fISIGMA\fP .PP .nf SIGMA is DOUBLE PRECISION The computed sigma_I, the I-th updated eigenvalue\&. .fi .PP .br \fIWORK\fP .PP .nf WORK is DOUBLE PRECISION array, dimension ( N ) If N \&.ne\&. 1, WORK contains (D(j) + sigma_I) in its j-th component\&. If N = 1, then WORK( 1 ) = 1\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit > 0: if INFO = 1, the updating process failed\&. .fi .PP .RE .PP \fBInternal Parameters:\fP .RS 4 .PP .nf Logical variable ORGATI (origin-at-i?) is used for distinguishing whether D(i) or D(i+1) is treated as the origin\&. ORGATI = \&.true\&. origin at i ORGATI = \&.false\&. origin at i+1 Logical variable SWTCH3 (switch-for-3-poles?) is for noting if we are working with THREE poles! MAXIT is the maximum number of iterations allowed for each eigenvalue\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBContributors:\fP .RS 4 Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA .RE .PP .SS "subroutine slasd4 (integer n, integer i, real, dimension( * ) d, real, dimension( * ) z, real, dimension( * ) delta, real rho, real sigma, real, dimension( * ) work, integer info)" .PP \fBSLASD4\fP computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix\&. Used by sbdsdc\&. .PP \fBPurpose:\fP .RS 4 .PP .nf This subroutine computes the square root of the I-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix whose entries are given as the squares of the corresponding entries in the array d, and that 0 <= D(i) < D(j) for i < j and that RHO > 0\&. This is arranged by the calling routine, and is no loss in generality\&. The rank-one modified system is thus diag( D ) * diag( D ) + RHO * Z * Z_transpose\&. where we assume the Euclidean norm of Z is 1\&. The method consists of approximating the rational functions in the secular equation by simpler interpolating rational functions\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIN\fP .PP .nf N is INTEGER The length of all arrays\&. .fi .PP .br \fII\fP .PP .nf I is INTEGER The index of the eigenvalue to be computed\&. 1 <= I <= N\&. .fi .PP .br \fID\fP .PP .nf D is REAL array, dimension ( N ) The original eigenvalues\&. It is assumed that they are in order, 0 <= D(I) < D(J) for I < J\&. .fi .PP .br \fIZ\fP .PP .nf Z is REAL array, dimension ( N ) The components of the updating vector\&. .fi .PP .br \fIDELTA\fP .PP .nf DELTA is REAL array, dimension ( N ) If N \&.ne\&. 1, DELTA contains (D(j) - sigma_I) in its j-th component\&. If N = 1, then DELTA(1) = 1\&. The vector DELTA contains the information necessary to construct the (singular) eigenvectors\&. .fi .PP .br \fIRHO\fP .PP .nf RHO is REAL The scalar in the symmetric updating formula\&. .fi .PP .br \fISIGMA\fP .PP .nf SIGMA is REAL The computed sigma_I, the I-th updated eigenvalue\&. .fi .PP .br \fIWORK\fP .PP .nf WORK is REAL array, dimension ( N ) If N \&.ne\&. 1, WORK contains (D(j) + sigma_I) in its j-th component\&. If N = 1, then WORK( 1 ) = 1\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit > 0: if INFO = 1, the updating process failed\&. .fi .PP .RE .PP \fBInternal Parameters:\fP .RS 4 .PP .nf Logical variable ORGATI (origin-at-i?) is used for distinguishing whether D(i) or D(i+1) is treated as the origin\&. ORGATI = \&.true\&. origin at i ORGATI = \&.false\&. origin at i+1 Logical variable SWTCH3 (switch-for-3-poles?) is for noting if we are working with THREE poles! MAXIT is the maximum number of iterations allowed for each eigenvalue\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBContributors:\fP .RS 4 Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA .RE .PP .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.