# laed7 - Man Page

laed7: D&C step: merge subproblems

## Synopsis

### Functions

subroutine claed7 (n, cutpnt, qsiz, tlvls, curlvl, curpbm, d, q, ldq, rho, indxq, qstore, qptr, prmptr, perm, givptr, givcol, givnum, work, rwork, iwork, info)
CLAED7 used by CSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.
subroutine dlaed7 (icompq, n, qsiz, tlvls, curlvl, curpbm, d, q, ldq, indxq, rho, cutpnt, qstore, qptr, prmptr, perm, givptr, givcol, givnum, work, iwork, info)
DLAED7 used by DSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.
subroutine slaed7 (icompq, n, qsiz, tlvls, curlvl, curpbm, d, q, ldq, indxq, rho, cutpnt, qstore, qptr, prmptr, perm, givptr, givcol, givnum, work, iwork, info)
SLAED7 used by SSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.
subroutine zlaed7 (n, cutpnt, qsiz, tlvls, curlvl, curpbm, d, q, ldq, rho, indxq, qstore, qptr, prmptr, perm, givptr, givcol, givnum, work, rwork, iwork, info)
ZLAED7 used by ZSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.

## Function Documentation

### subroutine claed7 (integer n, integer cutpnt, integer qsiz, integer tlvls, integer curlvl, integer curpbm, real, dimension( * ) d, complex, dimension( ldq, * ) q, integer ldq, real rho, integer, dimension( * ) indxq, real, dimension( * ) qstore, integer, dimension( * ) qptr, integer, dimension( * ) prmptr, integer, dimension( * ) perm, integer, dimension( * ) givptr, integer, dimension( 2, * ) givcol, real, dimension( 2, * ) givnum, complex, dimension( * ) work, real, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)

CLAED7 used by CSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.

Purpose:

``` CLAED7 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 optionally eigenvectors of a dense or banded
Hermitian matrix that has been reduced to tridiagonal form.

T = Q(in) ( D(in) + RHO * Z*Z**H ) Q**H(in) = Q(out) * D(out) * Q**H(out)

where Z = Q**Hu, 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 occurrence 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.```
Parameters

N

```          N is INTEGER
The dimension of the symmetric tridiagonal matrix.  N >= 0.```

CUTPNT

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

QSIZ

```          QSIZ is INTEGER
The dimension of the unitary matrix used to reduce
the full matrix to tridiagonal form.  QSIZ >= N.```

TLVLS

```          TLVLS is INTEGER
The total number of merging levels in the overall divide and
conquer tree.```

CURLVL

```          CURLVL is INTEGER
The current level in the overall merge routine,
0 <= curlvl <= tlvls.```

CURPBM

```          CURPBM is INTEGER
The current problem in the current level in the overall
merge routine (counting from upper left to lower right).```

D

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

Q

```          Q is COMPLEX array, dimension (LDQ,N)
On entry, the eigenvectors of the rank-1-perturbed matrix.
On exit, the eigenvectors of the repaired tridiagonal matrix.```

LDQ

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

RHO

```          RHO is REAL
Contains the subdiagonal element used to create the rank-1
modification.```

INDXQ

```          INDXQ is INTEGER array, dimension (N)
This contains the permutation which will reintegrate the
subproblem just solved back into sorted order,
ie. D( INDXQ( I = 1, N ) ) will be in ascending order.```

IWORK

`          IWORK is INTEGER array, dimension (4*N)`

RWORK

```          RWORK is REAL array,
dimension (3*N+2*QSIZ*N)```

WORK

`          WORK is COMPLEX array, dimension (QSIZ*N)`

QSTORE

```          QSTORE is REAL array, dimension (N**2+1)
Stores eigenvectors of submatrices encountered during
divide and conquer, packed together. QPTR points to
beginning of the submatrices.```

QPTR

```          QPTR is INTEGER array, dimension (N+2)
List of indices pointing to beginning of submatrices stored
in QSTORE. The submatrices are numbered starting at the
bottom left of the divide and conquer tree, from left to
right and bottom to top.```

PRMPTR

```          PRMPTR is INTEGER array, dimension (N lg N)
Contains a list of pointers which indicate where in PERM a
level's permutation is stored.  PRMPTR(i+1) - PRMPTR(i)
indicates the size of the permutation and also the size of
the full, non-deflated problem.```

PERM

```          PERM is INTEGER array, dimension (N lg N)
Contains the permutations (from deflation and sorting) to be
applied to each eigenblock.```

GIVPTR

```          GIVPTR is INTEGER array, dimension (N lg N)
Contains a list of pointers which indicate where in GIVCOL a
level's Givens rotations are stored.  GIVPTR(i+1) - GIVPTR(i)
indicates the number of Givens rotations.```

GIVCOL

```          GIVCOL is INTEGER array, dimension (2, N lg N)
Each pair of numbers indicates a pair of columns to take place
in a Givens rotation.```

GIVNUM

```          GIVNUM is REAL array, dimension (2, N lg N)
Each number indicates the S value to be used in the
corresponding Givens rotation.```

INFO

```          INFO is 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```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Definition at line 245 of file claed7.f.

### subroutine dlaed7 (integer icompq, integer n, integer qsiz, integer tlvls, integer curlvl, integer curpbm, double precision, dimension( * ) d, double precision, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) indxq, double precision rho, integer cutpnt, double precision, dimension( * ) qstore, integer, dimension( * ) qptr, integer, dimension( * ) prmptr, integer, dimension( * ) perm, integer, dimension( * ) givptr, integer, dimension( 2, * ) givcol, double precision, dimension( 2, * ) givnum, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DLAED7 used by DSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.

Purpose:

``` DLAED7 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 optionally eigenvectors of a dense symmetric matrix
that has been reduced to tridiagonal form.  DLAED1 handles
the case in which all eigenvalues and eigenvectors of a symmetric
tridiagonal matrix are desired.

T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)

where Z = Q**Tu, 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 occurrence the dimension of the
secular equation problem is reduced by one.  This stage is
performed by the routine DLAED8.

The second stage consists of calculating the updated
eigenvalues. This is done by finding the roots of the secular
equation via the routine DLAED4 (as called by DLAED9).
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.```
Parameters

ICOMPQ

```          ICOMPQ is INTEGER
= 0:  Compute eigenvalues only.
= 1:  Compute eigenvectors of original dense symmetric matrix
also.  On entry, Q contains the orthogonal matrix used
to reduce the original matrix to tridiagonal form.```

N

```          N is INTEGER
The dimension of the symmetric tridiagonal matrix.  N >= 0.```

QSIZ

```          QSIZ is INTEGER
The dimension of the orthogonal matrix used to reduce
the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.```

TLVLS

```          TLVLS is INTEGER
The total number of merging levels in the overall divide and
conquer tree.```

CURLVL

```          CURLVL is INTEGER
The current level in the overall merge routine,
0 <= CURLVL <= TLVLS.```

CURPBM

```          CURPBM is INTEGER
The current problem in the current level in the overall
merge routine (counting from upper left to lower right).```

D

```          D is DOUBLE PRECISION array, dimension (N)
On entry, the eigenvalues of the rank-1-perturbed matrix.
On exit, the eigenvalues of the repaired matrix.```

Q

```          Q is DOUBLE PRECISION array, dimension (LDQ, N)
On entry, the eigenvectors of the rank-1-perturbed matrix.
On exit, the eigenvectors of the repaired tridiagonal matrix.```

LDQ

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

INDXQ

```          INDXQ is INTEGER array, dimension (N)
The permutation which will reintegrate the subproblem just
solved back into sorted order, i.e., D( INDXQ( I = 1, N ) )
will be in ascending order.```

RHO

```          RHO is DOUBLE PRECISION
The subdiagonal element used to create the rank-1
modification.```

CUTPNT

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

QSTORE

```          QSTORE is DOUBLE PRECISION array, dimension (N**2+1)
Stores eigenvectors of submatrices encountered during
divide and conquer, packed together. QPTR points to
beginning of the submatrices.```

QPTR

```          QPTR is INTEGER array, dimension (N+2)
List of indices pointing to beginning of submatrices stored
in QSTORE. The submatrices are numbered starting at the
bottom left of the divide and conquer tree, from left to
right and bottom to top.```

PRMPTR

```          PRMPTR is INTEGER array, dimension (N lg N)
Contains a list of pointers which indicate where in PERM a
level's permutation is stored.  PRMPTR(i+1) - PRMPTR(i)
indicates the size of the permutation and also the size of
the full, non-deflated problem.```

PERM

```          PERM is INTEGER array, dimension (N lg N)
Contains the permutations (from deflation and sorting) to be
applied to each eigenblock.```

GIVPTR

```          GIVPTR is INTEGER array, dimension (N lg N)
Contains a list of pointers which indicate where in GIVCOL a
level's Givens rotations are stored.  GIVPTR(i+1) - GIVPTR(i)
indicates the number of Givens rotations.```

GIVCOL

```          GIVCOL is INTEGER array, dimension (2, N lg N)
Each pair of numbers indicates a pair of columns to take place
in a Givens rotation.```

GIVNUM

```          GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N)
Each number indicates the S value to be used in the
corresponding Givens rotation.```

WORK

`          WORK is DOUBLE PRECISION array, dimension (3*N+2*QSIZ*N)`

IWORK

`          IWORK is INTEGER array, dimension (4*N)`

INFO

```          INFO is 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```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

Definition at line 256 of file dlaed7.f.

### subroutine slaed7 (integer icompq, integer n, integer qsiz, integer tlvls, integer curlvl, integer curpbm, real, dimension( * ) d, real, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) indxq, real rho, integer cutpnt, real, dimension( * ) qstore, integer, dimension( * ) qptr, integer, dimension( * ) prmptr, integer, dimension( * ) perm, integer, dimension( * ) givptr, integer, dimension( 2, * ) givcol, real, dimension( 2, * ) givnum, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SLAED7 used by SSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.

Purpose:

``` SLAED7 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 optionally eigenvectors of a dense symmetric matrix
that has been reduced to tridiagonal form.  SLAED1 handles
the case in which all eigenvalues and eigenvectors of a symmetric
tridiagonal matrix are desired.

T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)

where Z = Q**Tu, 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 occurrence the dimension of the
secular equation problem is reduced by one.  This stage is
performed by the routine SLAED8.

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 SLAED9).
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.```
Parameters

ICOMPQ

```          ICOMPQ is INTEGER
= 0:  Compute eigenvalues only.
= 1:  Compute eigenvectors of original dense symmetric matrix
also.  On entry, Q contains the orthogonal matrix used
to reduce the original matrix to tridiagonal form.```

N

```          N is INTEGER
The dimension of the symmetric tridiagonal matrix.  N >= 0.```

QSIZ

```          QSIZ is INTEGER
The dimension of the orthogonal matrix used to reduce
the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.```

TLVLS

```          TLVLS is INTEGER
The total number of merging levels in the overall divide and
conquer tree.```

CURLVL

```          CURLVL is INTEGER
The current level in the overall merge routine,
0 <= CURLVL <= TLVLS.```

CURPBM

```          CURPBM is INTEGER
The current problem in the current level in the overall
merge routine (counting from upper left to lower right).```

D

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

Q

```          Q is 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

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

INDXQ

```          INDXQ is INTEGER array, dimension (N)
The permutation which will reintegrate the subproblem just
solved back into sorted order, i.e., D( INDXQ( I = 1, N ) )
will be in ascending order.```

RHO

```          RHO is REAL
The subdiagonal element used to create the rank-1
modification.```

CUTPNT

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

QSTORE

```          QSTORE is REAL array, dimension (N**2+1)
Stores eigenvectors of submatrices encountered during
divide and conquer, packed together. QPTR points to
beginning of the submatrices.```

QPTR

```          QPTR is INTEGER array, dimension (N+2)
List of indices pointing to beginning of submatrices stored
in QSTORE. The submatrices are numbered starting at the
bottom left of the divide and conquer tree, from left to
right and bottom to top.```

PRMPTR

```          PRMPTR is INTEGER array, dimension (N lg N)
Contains a list of pointers which indicate where in PERM a
level's permutation is stored.  PRMPTR(i+1) - PRMPTR(i)
indicates the size of the permutation and also the size of
the full, non-deflated problem.```

PERM

```          PERM is INTEGER array, dimension (N lg N)
Contains the permutations (from deflation and sorting) to be
applied to each eigenblock.```

GIVPTR

```          GIVPTR is INTEGER array, dimension (N lg N)
Contains a list of pointers which indicate where in GIVCOL a
level's Givens rotations are stored.  GIVPTR(i+1) - GIVPTR(i)
indicates the number of Givens rotations.```

GIVCOL

```          GIVCOL is INTEGER array, dimension (2, N lg N)
Each pair of numbers indicates a pair of columns to take place
in a Givens rotation.```

GIVNUM

```          GIVNUM is REAL array, dimension (2, N lg N)
Each number indicates the S value to be used in the
corresponding Givens rotation.```

WORK

`          WORK is REAL array, dimension (3*N+2*QSIZ*N)`

IWORK

`          IWORK is INTEGER array, dimension (4*N)`

INFO

```          INFO is 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```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

Definition at line 256 of file slaed7.f.

### subroutine zlaed7 (integer n, integer cutpnt, integer qsiz, integer tlvls, integer curlvl, integer curpbm, double precision, dimension( * ) d, complex*16, dimension( ldq, * ) q, integer ldq, double precision rho, integer, dimension( * ) indxq, double precision, dimension( * ) qstore, integer, dimension( * ) qptr, integer, dimension( * ) prmptr, integer, dimension( * ) perm, integer, dimension( * ) givptr, integer, dimension( 2, * ) givcol, double precision, dimension( 2, * ) givnum, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)

ZLAED7 used by ZSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.

Purpose:

``` ZLAED7 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 optionally eigenvectors of a dense or banded
Hermitian matrix that has been reduced to tridiagonal form.

T = Q(in) ( D(in) + RHO * Z*Z**H ) Q**H(in) = Q(out) * D(out) * Q**H(out)

where Z = Q**Hu, 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 occurrence the dimension of the
secular equation problem is reduced by one.  This stage is
performed by the routine DLAED2.

The second stage consists of calculating the updated
eigenvalues. This is done by finding the roots of the secular
equation via the routine DLAED4 (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.```
Parameters

N

```          N is INTEGER
The dimension of the symmetric tridiagonal matrix.  N >= 0.```

CUTPNT

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

QSIZ

```          QSIZ is INTEGER
The dimension of the unitary matrix used to reduce
the full matrix to tridiagonal form.  QSIZ >= N.```

TLVLS

```          TLVLS is INTEGER
The total number of merging levels in the overall divide and
conquer tree.```

CURLVL

```          CURLVL is INTEGER
The current level in the overall merge routine,
0 <= curlvl <= tlvls.```

CURPBM

```          CURPBM is INTEGER
The current problem in the current level in the overall
merge routine (counting from upper left to lower right).```

D

```          D is DOUBLE PRECISION array, dimension (N)
On entry, the eigenvalues of the rank-1-perturbed matrix.
On exit, the eigenvalues of the repaired matrix.```

Q

```          Q is COMPLEX*16 array, dimension (LDQ,N)
On entry, the eigenvectors of the rank-1-perturbed matrix.
On exit, the eigenvectors of the repaired tridiagonal matrix.```

LDQ

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

RHO

```          RHO is DOUBLE PRECISION
Contains the subdiagonal element used to create the rank-1
modification.```

INDXQ

```          INDXQ is INTEGER array, dimension (N)
This contains the permutation which will reintegrate the
subproblem just solved back into sorted order,
ie. D( INDXQ( I = 1, N ) ) will be in ascending order.```

IWORK

`          IWORK is INTEGER array, dimension (4*N)`

RWORK

```          RWORK is DOUBLE PRECISION array,
dimension (3*N+2*QSIZ*N)```

WORK

`          WORK is COMPLEX*16 array, dimension (QSIZ*N)`

QSTORE

```          QSTORE is DOUBLE PRECISION array, dimension (N**2+1)
Stores eigenvectors of submatrices encountered during
divide and conquer, packed together. QPTR points to
beginning of the submatrices.```

QPTR

```          QPTR is INTEGER array, dimension (N+2)
List of indices pointing to beginning of submatrices stored
in QSTORE. The submatrices are numbered starting at the
bottom left of the divide and conquer tree, from left to
right and bottom to top.```

PRMPTR

```          PRMPTR is INTEGER array, dimension (N lg N)
Contains a list of pointers which indicate where in PERM a
level's permutation is stored.  PRMPTR(i+1) - PRMPTR(i)
indicates the size of the permutation and also the size of
the full, non-deflated problem.```

PERM

```          PERM is INTEGER array, dimension (N lg N)
Contains the permutations (from deflation and sorting) to be
applied to each eigenblock.```

GIVPTR

```          GIVPTR is INTEGER array, dimension (N lg N)
Contains a list of pointers which indicate where in GIVCOL a
level's Givens rotations are stored.  GIVPTR(i+1) - GIVPTR(i)
indicates the number of Givens rotations.```

GIVCOL

```          GIVCOL is INTEGER array, dimension (2, N lg N)
Each pair of numbers indicates a pair of columns to take place
in a Givens rotation.```

GIVNUM

```          GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N)
Each number indicates the S value to be used in the
corresponding Givens rotation.```

INFO

```          INFO is 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```
Author

Univ. of Tennessee

Univ. of California Berkeley