Thread Subject: Questions on matrix operations (inverse) and mvnrnd

Hello Matlab users,

I have questions on matrix operations and appreciate some
help.

First, I need to inverse an n by n symmetric matrix H,
which is certainly not singular. Since I would receive a
warning message as 'Matrix is close to singular or badly
scaled', if I would use 'inv(H)'. Thus I have tried four
alternatives respectively as follows.
(A) Hinv = H\eye(n);
(B) Hinv = linsolve(H, eye(n));
(C) temp = chol(H)\eye(n); Hinv=temp*temp';
(D) Hinv = pinv(H).
I still received warning messages on singularity in the
first two cases (A) and (B). Although I did not receive any
warning messages in the last two cases (C) and (D), the
results do not appear to be accurate enough to pass some
tests.

Second, I need to use 'mvnrnd(MU, SIGMA)' to obtain random
vectors. The covariance matrix SIGMA in my case is
specified as the inverse of H denoted as Hinv. Although
Hinv looks like a symmetric matrix in an array editor,
Matlab returns an error message as 'SIGMA must be a
symmetric positive semi-definite matrix'. Thus I have tried
an althernative:
sigma = Hinv-(Hinv-Hinv')./2.
Then, I will not receive error messages.

In short, I wanted to know how to effectively
(1) obtian the inverse of an n by n matrix H (denote it as
Hinv);
(2) use mvnrnd(MU, SIGMA) by setting SIGMA equal to Hiv;
without receiving warning/error messages.

Thank you very much!

Ycshmily.

Ycshmily M wrote:

> First, I need to inverse an n by n symmetric matrix H,
> which is certainly not singular. Since I would receive a
> warning message as 'Matrix is close to singular or badly
> scaled'

Ycshmily, apparently H _is_ singular, or at least it is "numerically singular",
meaning that it is so close to being singular that any numerical operations
(such as inversion) are doomed to be unstable.

, if I would use 'inv(H)'. Thus I have tried four
> alternatives respectively as follows.
> (A) Hinv = H\eye(n);
> (B) Hinv = linsolve(H, eye(n));
> (C) temp = chol(H)\eye(n); Hinv=temp*temp';
> (D) Hinv = pinv(H).
> I still received warning messages on singularity in the
> first two cases (A) and (B). Although I did not receive any
> warning messages in the last two cases (C) and (D), the
> results do not appear to be accurate enough to pass some
> tests.
>
> Second, I need to use 'mvnrnd(MU, SIGMA)' to obtain random
> vectors. The covariance matrix SIGMA in my case is
> specified as the inverse of H denoted as Hinv. Although
> Hinv looks like a symmetric matrix in an array editor,
> Matlab returns an error message as 'SIGMA must be a
> symmetric positive semi-definite matrix'.

You may be giving MVNRND a matrix that is not symmetric, but based on your
earlier comments, it may also be that it is not (numerically) positive
semi-definite. Set a break point in MVNRND (or CHOLCOV) and find out why it is
erroring out.

Hope this helps.

Hello Peter,

Thanks for your reply!

In my case, the H matrix is a large (could be 400 by 400)
sparse matrix, which I suspect could be a reason for those
warning/error messages because the way Matlab handles
sparse matrices. Therefore, I am looking for some effective
transformations to circumvent the problem, like using
Cholesky decomposition at first and then taking the
inverse...

Ycshmily.

Peter Perkins <Peter.PerkinsRemoveThis@mathworks.com> wrote
in message <g5nf6u$15b$1@fred.mathworks.com>...
> Ycshmily M wrote:
>
> > First, I need to inverse an n by n symmetric matrix H,
> > which is certainly not singular. Since I would receive
a
> > warning message as 'Matrix is close to singular or
badly
> > scaled'
>
> Ycshmily, apparently H _is_ singular, or at least it
is "numerically singular",
> meaning that it is so close to being singular that any
numerical operations
> (such as inversion) are doomed to be unstable.
>
> , if I would use 'inv(H)'. Thus I have tried four
> > alternatives respectively as follows.
> > (A) Hinv = H\eye(n);
> > (B) Hinv = linsolve(H, eye(n));
> > (C) temp = chol(H)\eye(n); Hinv=temp*temp';
> > (D) Hinv = pinv(H).
> > I still received warning messages on singularity in the
> > first two cases (A) and (B). Although I did not receive
any
> > warning messages in the last two cases (C) and (D), the
> > results do not appear to be accurate enough to pass
some
> > tests.
> >
> > Second, I need to use 'mvnrnd(MU, SIGMA)' to obtain
random
> > vectors. The covariance matrix SIGMA in my case is
> > specified as the inverse of H denoted as Hinv. Although
> > Hinv looks like a symmetric matrix in an array editor,
> > Matlab returns an error message as 'SIGMA must be a
> > symmetric positive semi-definite matrix'.
>
> You may be giving MVNRND a matrix that is not symmetric,
but based on your
> earlier comments, it may also be that it is not
(numerically) positive
> semi-definite. Set a break point in MVNRND (or CHOLCOV)
and find out why it is
> erroring out.
>
> Hope this helps.

"Ycshmily M" <ycshmily@yahoo.com> wrote in message
<g5nmju$64o$1@fred.mathworks.com>...
> Hello Peter,
>
> Thanks for your reply!
>
> In my case, the H matrix is a large (could be 400 by 400)
> sparse matrix, which I suspect could be a reason for those
> warning/error messages because the way Matlab handles
> sparse matrices. Therefore, I am looking for some effective
> transformations to circumvent the problem, like using
> Cholesky decomposition at first and then taking the
> inverse...
>
> Ycshmily.

That's what inv does in the sparse case, I think.

Do you really truly need the inverse? What are you doing
with the inverse?

You might also try pinv(full(A)), since A is small (only
400-by-400). The pseudo inverse always exists, whether or
not A is singular.

Ycshmily M wrote:

> First, I need to inverse an n by n symmetric matrix H,
> which is certainly not singular. Since I would receive a
> warning message as 'Matrix is close to singular or badly
> scaled'

Ycshmily, apparently H _is_ singular, or at least it is "numerically singular",
meaning that it is so close to being singular that any numerical operations
(such as inversion) are doomed to be unstable.

, if I would use 'inv(H)'. Thus I have tried four
> alternatives respectively as follows.
> (A) Hinv = H\eye(n);
> (B) Hinv = linsolve(H, eye(n));
> (C) temp = chol(H)\eye(n); Hinv=temp*temp';
> (D) Hinv = pinv(H).
> I still received warning messages on singularity in the
> first two cases (A) and (B). Although I did not receive any
> warning messages in the last two cases (C) and (D), the
> results do not appear to be accurate enough to pass some
> tests.
>
> Second, I need to use 'mvnrnd(MU, SIGMA)' to obtain random
> vectors. The covariance matrix SIGMA in my case is
> specified as the inverse of H denoted as Hinv. Although
> Hinv looks like a symmetric matrix in an array editor,
> Matlab returns an error message as 'SIGMA must be a
> symmetric positive semi-definite matrix'.

You may be giving MVNRND a matrix that is not symmetric, but based on your
earlier comments, it may also be that it is not (numerically) positive
semi-definite. Set a break point in MVNRND (or CHOLCOV) and find out why it is
erroring out.

Hope this helps.

Hello Peter,

Thanks for your reply!

In my case, the H matrix is a large (could be 400 by 400)
sparse matrix, which I suspect could be a reason for those
warning/error messages because the way Matlab handles
sparse matrices. Therefore, I am looking for some effective
transformations to circumvent the problem, like using
Cholesky decomposition at first and then taking the
inverse...

Ycshmily.

Peter Perkins <Peter.PerkinsRemoveThis@mathworks.com> wrote
in message <g5nf6u$15b$1@fred.mathworks.com>...
> Ycshmily M wrote:
>
> > First, I need to inverse an n by n symmetric matrix H,
> > which is certainly not singular. Since I would receive
a
> > warning message as 'Matrix is close to singular or
badly
> > scaled'
>
> Ycshmily, apparently H _is_ singular, or at least it
is "numerically singular",
> meaning that it is so close to being singular that any
numerical operations
> (such as inversion) are doomed to be unstable.
>
> , if I would use 'inv(H)'. Thus I have tried four
> > alternatives respectively as follows.
> > (A) Hinv = H\eye(n);
> > (B) Hinv = linsolve(H, eye(n));
> > (C) temp = chol(H)\eye(n); Hinv=temp*temp';
> > (D) Hinv = pinv(H).
> > I still received warning messages on singularity in the
> > first two cases (A) and (B). Although I did not receive
any
> > warning messages in the last two cases (C) and (D), the
> > results do not appear to be accurate enough to pass
some
> > tests.
> >
> > Second, I need to use 'mvnrnd(MU, SIGMA)' to obtain
random
> > vectors. The covariance matrix SIGMA in my case is
> > specified as the inverse of H denoted as Hinv. Although
> > Hinv looks like a symmetric matrix in an array editor,
> > Matlab returns an error message as 'SIGMA must be a
> > symmetric positive semi-definite matrix'.
>
> You may be giving MVNRND a matrix that is not symmetric,
but based on your
> earlier comments, it may also be that it is not
(numerically) positive
> semi-definite. Set a break point in MVNRND (or CHOLCOV)
and find out why it is
> erroring out.
>
> Hope this helps.

"Ycshmily M" <ycshmily@yahoo.com> wrote in message
<g5nmju$64o$1@fred.mathworks.com>...
> Hello Peter,
>
> Thanks for your reply!
>
> In my case, the H matrix is a large (could be 400 by 400)
> sparse matrix, which I suspect could be a reason for those
> warning/error messages because the way Matlab handles
> sparse matrices. Therefore, I am looking for some effective
> transformations to circumvent the problem, like using
> Cholesky decomposition at first and then taking the
> inverse...
>
> Ycshmily.

That's what inv does in the sparse case, I think.

Do you really truly need the inverse? What are you doing
with the inverse?

You might also try pinv(full(A)), since A is small (only
400-by-400). The pseudo inverse always exists, whether or
not A is singular.