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man pages->IRIX man pages -> complib/slaed1 (3)
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### Contents

```
SLAED1(3F)							    SLAED1(3F)

```

### NAME[Toc][Back]

```     SLAED1 - compute the updated eigensystem of a diagonal matrix after
modification by a rank-one	symmetric matrix
```

### SYNOPSIS[Toc][Back]

```     SUBROUTINE	SLAED1(	N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK,	IWORK, INFO )

INTEGER	CUTPNT,	INFO, LDQ, N

REAL		RHO

INTEGER	INDXQ( * ), IWORK( * )

REAL		D( * ),	Q( LDQ,	* ), WORK( * )
```

### PURPOSE[Toc][Back]

```     SLAED1 computes the updated eigensystem of	a diagonal matrix after
modification by a rank-one	symmetric matrix.  This	routine	is used	only
for the eigenproblem which	requires all eigenvalues and eigenvectors of a
tridiagonal matrix.  SLAED7 handles the case in which eigenvalues only or
eigenvalues and eigenvectors of a full symmetric matrix (which was
reduced to	tridiagonal form) are desired.

T = Q(in) ( D(in) + RHO * Z*Z' )	Q'(in) = Q(out)	* D(out) * Q'(out)

where Z	= Q'u, u is a vector of	length N with ones in the
CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.

The eigenvectors of the	original matrix	are stored in Q, and the
eigenvalues are	in D.  The algorithm consists of three stages:

The first stage consists of deflating the size of the problem
when	there are multiple eigenvalues or if there is a	zero in
the Z vector.  For each such	occurence the dimension	of the
secular equation problem is reduced by one.	This stage is
performed by	the routine SLAED2.

The second stage consists of	calculating the	updated
eigenvalues.	This is	done by	finding	the roots of the secular
equation via	the routine SLAED4 (as called by SLAED3).
This	routine	also calculates	the eigenvectors of the	current
problem.

The final stage consists of computing the updated eigenvectors
directly using the updated eigenvalues.  The	eigenvectors for
the current problem are multiplied with the eigenvectors from
the overall problem.

Page 1

SLAED1(3F)							    SLAED1(3F)

ARGUMENTS
N	    (input) INTEGER
The	dimension of the symmetric tridiagonal matrix.	N >= 0.

D	    (input/output) REAL	array, dimension (N)
On entry, the eigenvalues of the rank-1-perturbed matrix.  On
exit, the eigenvalues of the repaired matrix.

Q	    (input/output) REAL	array, dimension (LDQ,N)
On entry, the eigenvectors of the rank-1-perturbed matrix.	On
exit, the eigenvectors of the repaired tridiagonal matrix.

LDQ    (input) INTEGER
The	leading	dimension of the array Q.  LDQ >= max(1,N).

INDXQ  (input/output) INTEGER array, dimension (N)
On entry, the permutation which separately sorts the two
subproblems	in D into ascending order.  On exit, the permutation
which will reintegrate the subproblems back	into sorted order,
i.e. D( INDXQ( I = 1, N ) )	will be	in ascending order.

RHO    (input) REAL
The	subdiagonal entry used to create the rank-1 modification.

CUTPNT (input) INTEGER The location	of the last eigenvalue in the
leading sub-matrix.	 min(1,N) <= CUTPNT <= N.

WORK   (workspace)	REAL array, dimension (3*N+2*N**2)

IWORK  (workspace)	INTEGER	array, dimension (4*N)

INFO   (output) INTEGER
= 0:  successful exit.
< 0:  if INFO = -i,	the i-th argument had an illegal value.
> 0:  if INFO = 1, an eigenvalue did not converge
SLAED1(3F)							    SLAED1(3F)

```

### NAME[Toc][Back]

```     SLAED1 - compute the updated eigensystem of a diagonal matrix after
modification by a rank-one	symmetric matrix
```

### SYNOPSIS[Toc][Back]

```     SUBROUTINE	SLAED1(	N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK,	IWORK, INFO )

INTEGER	CUTPNT,	INFO, LDQ, N

REAL		RHO

INTEGER	INDXQ( * ), IWORK( * )

REAL		D( * ),	Q( LDQ,	* ), WORK( * )
```

### PURPOSE[Toc][Back]

```     SLAED1 computes the updated eigensystem of	a diagonal matrix after
modification by a rank-one	symmetric matrix.  This	routine	is used	only
for the eigenproblem which	requires all eigenvalues and eigenvectors of a
tridiagonal matrix.  SLAED7 handles the case in which eigenvalues only or
eigenvalues and eigenvectors of a full symmetric matrix (which was
reduced to	tridiagonal form) are desired.

T = Q(in) ( D(in) + RHO * Z*Z' )	Q'(in) = Q(out)	* D(out) * Q'(out)

where Z	= Q'u, u is a vector of	length N with ones in the
CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.

The eigenvectors of the	original matrix	are stored in Q, and the
eigenvalues are	in D.  The algorithm consists of three stages:

The first stage consists of deflating the size of the problem
when	there are multiple eigenvalues or if there is a	zero in
the Z vector.  For each such	occurence the dimension	of the
secular equation problem is reduced by one.	This stage is
performed by	the routine SLAED2.

The second stage consists of	calculating the	updated
eigenvalues.	This is	done by	finding	the roots of the secular
equation via	the routine SLAED4 (as called by SLAED3).
This	routine	also calculates	the eigenvectors of the	current
problem.

The final stage consists of computing the updated eigenvectors
directly using the updated eigenvalues.  The	eigenvectors for
the current problem are multiplied with the eigenvectors from
the overall problem.

Page 1

SLAED1(3F)							    SLAED1(3F)

ARGUMENTS
N	    (input) INTEGER
The	dimension of the symmetric tridiagonal matrix.	N >= 0.

D	    (input/output) REAL	array, dimension (N)
On entry, the eigenvalues of the rank-1-perturbed matrix.  On
exit, the eigenvalues of the repaired matrix.

Q	    (input/output) REAL	array, dimension (LDQ,N)
On entry, the eigenvectors of the rank-1-perturbed matrix.	On
exit, the eigenvectors of the repaired tridiagonal matrix.

LDQ    (input) INTEGER
The	leading	dimension of the array Q.  LDQ >= max(1,N).

INDXQ  (input/output) INTEGER array, dimension (N)
On entry, the permutation which separately sorts the two
subproblems	in D into ascending order.  On exit, the permutation
which will reintegrate the subproblems back	into sorted order,
i.e. D( INDXQ( I = 1, N ) )	will be	in ascending order.

RHO    (input) REAL
The	subdiagonal entry used to create the rank-1 modification.

CUTPNT (input) INTEGER The location	of the last eigenvalue in the
leading sub-matrix.	 min(1,N) <= CUTPNT <= N.

WORK   (workspace)	REAL array, dimension (3*N+2*N**2)

IWORK  (workspace)	INTEGER	array, dimension (4*N)

INFO   (output) INTEGER
= 0:  successful exit.
< 0:  if INFO = -i,	the i-th argument had an illegal value.
> 0:  if INFO = 1, an eigenvalue did not converge

PPPPaaaaggggeeee 2222```
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