.TH "lasq1" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
lasq1 \- lasq1: dqds step
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBdlasq1\fP (n, d, e, work, info)"
.br
.RI "\fBDLASQ1\fP computes the singular values of a real square bidiagonal matrix\&. Used by sbdsqr\&. "
.ti -1c
.RI "subroutine \fBslasq1\fP (n, d, e, work, info)"
.br
.RI "\fBSLASQ1\fP computes the singular values of a real square bidiagonal matrix\&. Used by sbdsqr\&. "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "subroutine dlasq1 (integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( * ) work, integer info)"

.PP
\fBDLASQ1\fP computes the singular values of a real square bidiagonal matrix\&. Used by sbdsqr\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 DLASQ1 computes the singular values of a real N-by-N bidiagonal
 matrix with diagonal D and off-diagonal E\&. The singular values
 are computed to high relative accuracy, in the absence of
 denormalization, underflow and overflow\&. The algorithm was first
 presented in

 'Accurate singular values and differential qd algorithms' by K\&. V\&.
 Fernando and B\&. N\&. Parlett, Numer\&. Math\&., Vol-67, No\&. 2, pp\&. 191-230,
 1994,

 and the present implementation is described in 'An implementation of
 the dqds Algorithm (Positive Case)', LAPACK Working Note\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIN\fP 
.PP
.nf
          N is INTEGER
        The number of rows and columns in the matrix\&. N >= 0\&.
.fi
.PP
.br
\fID\fP 
.PP
.nf
          D is DOUBLE PRECISION array, dimension (N)
        On entry, D contains the diagonal elements of the
        bidiagonal matrix whose SVD is desired\&. On normal exit,
        D contains the singular values in decreasing order\&.
.fi
.PP
.br
\fIE\fP 
.PP
.nf
          E is DOUBLE PRECISION array, dimension (N)
        On entry, elements E(1:N-1) contain the off-diagonal elements
        of the bidiagonal matrix whose SVD is desired\&.
        On exit, E is overwritten\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is DOUBLE PRECISION array, dimension (4*N)
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          INFO is INTEGER
        = 0: successful exit
        < 0: if INFO = -i, the i-th argument had an illegal value
        > 0: the algorithm failed
             = 1, a split was marked by a positive value in E
             = 2, current block of Z not diagonalized after 100*N
                  iterations (in inner while loop)  On exit D and E
                  represent a matrix with the same singular values
                  which the calling subroutine could use to finish the
                  computation, or even feed back into DLASQ1
             = 3, termination criterion of outer while loop not met
                  (program created more than N unreduced blocks)
.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

.SS "subroutine slasq1 (integer n, real, dimension( * ) d, real, dimension( * ) e, real, dimension( * ) work, integer info)"

.PP
\fBSLASQ1\fP computes the singular values of a real square bidiagonal matrix\&. Used by sbdsqr\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 SLASQ1 computes the singular values of a real N-by-N bidiagonal
 matrix with diagonal D and off-diagonal E\&. The singular values
 are computed to high relative accuracy, in the absence of
 denormalization, underflow and overflow\&. The algorithm was first
 presented in

 'Accurate singular values and differential qd algorithms' by K\&. V\&.
 Fernando and B\&. N\&. Parlett, Numer\&. Math\&., Vol-67, No\&. 2, pp\&. 191-230,
 1994,

 and the present implementation is described in 'An implementation of
 the dqds Algorithm (Positive Case)', LAPACK Working Note\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIN\fP 
.PP
.nf
          N is INTEGER
        The number of rows and columns in the matrix\&. N >= 0\&.
.fi
.PP
.br
\fID\fP 
.PP
.nf
          D is REAL array, dimension (N)
        On entry, D contains the diagonal elements of the
        bidiagonal matrix whose SVD is desired\&. On normal exit,
        D contains the singular values in decreasing order\&.
.fi
.PP
.br
\fIE\fP 
.PP
.nf
          E is REAL array, dimension (N)
        On entry, elements E(1:N-1) contain the off-diagonal elements
        of the bidiagonal matrix whose SVD is desired\&.
        On exit, E is overwritten\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is REAL array, dimension (4*N)
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          INFO is INTEGER
        = 0: successful exit
        < 0: if INFO = -i, the i-th argument had an illegal value
        > 0: the algorithm failed
             = 1, a split was marked by a positive value in E
             = 2, current block of Z not diagonalized after 100*N
                  iterations (in inner while loop)  On exit D and E
                  represent a matrix with the same singular values
                  which the calling subroutine could use to finish the
                  computation, or even feed back into SLASQ1
             = 3, termination criterion of outer while loop not met
                  (program created more than N unreduced blocks)
.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

.SH "Author"
.PP 
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