Cholesky factorization
R = chol(A)
L = chol(A,'lower')
R
= chol(A,'upper')
[R,p] = chol(A)
[L,p] = chol(A,'lower')
[R,p] = chol(A,'upper')
[R,p,S] = chol(A)
[R,p,s] = chol(A,'vector')
[L,p,s] = chol(A,'lower','vector')
[R,p,s]
= chol(A,'upper','vector')
R = chol(A)
produces an
upper triangular matrix R
from the diagonal and
upper triangle of matrix A
, satisfying the equation R'*R=A
.
The chol
function assumes that A
is
(complex Hermitian) symmetric. If it is not, chol
uses
the (complex conjugate) transpose of the upper triangle as the lower
triangle. Matrix A
must be positive definite.
L = chol(A,'lower')
produces
a lower triangular matrix L
from the diagonal and
lower triangle of matrix A
, satisfying the equation L*L'=A
.
The chol
function assumes that A
is
(complex Hermitian) symmetric. If it is not, chol
uses
the (complex conjugate) transpose of the lower triangle as the upper
triangle. When A
is sparse, this syntax of chol
is
typically faster. Matrix A
must be positive definite. R
= chol(A,'upper')
is the same as R = chol(A)
.
[R,p] = chol(A)
for positive
definite A
, produces an upper triangular matrix R
from
the diagonal and upper triangle of matrix A
, satisfying
the equation R'*R=A
and p
is
zero. If A
is not positive definite, then p
is
a positive integer and MATLAB^{®} does not generate an error. When A
is
full, R
is an upper triangular matrix of order q=p-1
such
that R'*R=A(1:q,1:q)
. When A
is
sparse, R
is an upper triangular matrix of size q
-by-n
so
that the L
-shaped region of the first q
rows
and first q
columns of R'*R
agree
with those of A
.
[L,p] = chol(A,'lower')
for
positive definite A
, produces a lower triangular
matrix L
from the diagonal and lower triangle of
matrix A
, satisfying the equation L*L'=A
and p
is
zero. If A
is not positive definite, then p
is
a positive integer and MATLAB does not generate an error. When A
is
full, L
is a lower triangular matrix of order q=p-1
such
that L*L'=A(1:q,1:q)
. When A
is
sparse, L
is a lower triangular matrix of size q
-by-n
so
that the L
-shaped region of the first q
rows
and first q
columns of L*L'
agree
with those of A
. [R,p] = chol(A,'upper')
is
the same as [R,p] = chol(A)
.
The following three-output syntaxes require sparse input A
.
[R,p,S] = chol(A)
, when A
is
sparse, returns a permutation matrix S
. Note that
the preordering S
may differ from that obtained
from amd
since chol
will
slightly change the ordering for increased performance. When p=0
, R
is
an upper triangular matrix such that R'*R=S'*A*S
.
When p
is not zero, R
is an
upper triangular matrix of size q
-by-n
so
that the L
-shaped region of the first q
rows
and first q
columns of R'*R
agree
with those of S'*A*S
. The factor of S'*A*S
tends
to be sparser than the factor of A
.
[R,p,s] = chol(A,'vector')
,
when A
is sparse, returns the permutation information
as a vector s
such that A(s,s)=R'*R
,
when p=0
. You can use the 'matrix'
option
in place of 'vector'
to obtain the default behavior.
[L,p,s] = chol(A,'lower','vector')
,
when A
is sparse, uses only the diagonal and the
lower triangle of A
and returns a lower triangular
matrix L
and a permutation vector s
such
that A(s,s)=L*L'
, when p=0
.
As above, you can use the 'matrix'
option in place
of 'vector'
to obtain a permutation matrix. [R,p,s]
= chol(A,'upper','vector')
is the same as [R,p,s]
= chol(A,'vector')
.
Note
Using |
The gallery
function
provides several symmetric, positive, definite matrices.
A=gallery('moler',5) A = 1 -1 -1 -1 -1 -1 2 0 0 0 -1 0 3 1 1 -1 0 1 4 2 -1 0 1 2 5 C=chol(A) ans = 1 -1 -1 -1 -1 0 1 -1 -1 -1 0 0 1 -1 -1 0 0 0 1 -1 0 0 0 0 1 isequal(C'*C,A) ans = 1
For sparse input matrices, chol
returns
the Cholesky factor.
N = 100; A = gallery('poisson', N);
N
represents the number of grid points in
one direction of a square N
-by-N
grid.
Therefore, A
is $${\text{N}}^{2}$$ by $${\text{N}}^{2}$$.
L = chol(A, 'lower'); D = norm(A - L*L', 'fro');
The value of D
will vary somewhat among different
versions of MATLAB but will be on order of $${10}^{-14}$$.
The binomial coefficients arranged in a symmetric array create a positive definite matrix.
n = 5; X = pascal(n) X = 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
This matrix is interesting because its Cholesky factor consists of the same coefficients, arranged in an upper triangular matrix.
R = chol(X) R = 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1
Destroy the positive definiteness (and actually make the matrix singular) by subtracting 1 from the last element.
X(n,n) = X(n,n)-1 X = 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 69
Now an attempt to find the Cholesky factorization of X
fails.
chol(X) Error using chol Matrix must be positive definite.