Thread Subject: determinant of a banded matrix

Is it any easier to find the determinant of a banded matrix
than a full matrix? Any suggestions what's the best way to
do it?

"Deepak " <deepak.trivedi@gmail.com> wrote in message
<g3ei6q$6vo$1@fred.mathworks.com>...
> Is it any easier to find the determinant of a banded matrix
> than a full matrix? Any suggestions what's the best way to
> do it?
One simple solution (but there might be some simpler ones)
is to reduce the matrix to triangular form, as you will do
for Gaussian elimination. The determinant is just the
product of the diagonal elements.
That should cost of the O(n) instead of O(n^3) which is
typical for full matrix.
Regards
Carlos

"Deepak " <deepak.trivedi@gmail.com> wrote in message
<g3ei6q$6vo$1@fred.mathworks.com>...
> Is it any easier to find the determinant of a banded matrix
> than a full matrix? Any suggestions what's the best way to
> do it?

What do you mean by easier? DET(A) works like a charm for
both full and sparse matrices A.

Jos

"Deepak " <deepak.trivedi@gmail.com> wrote in message
<g3ei6q$6vo$1@fred.mathworks.com>...
> Is it any easier to find the determinant of a banded matrix
> than a full matrix? Any suggestions what's the best way to
> do it?

The more basic question is why? Why do you want to compute
the determinant? If you're worried about performance, then
that means you're trying to do this for large matrices. If
that's the case, then you really shouldn't use det at all.

Det is useless for practical computations. It's only of
theoretical importance. Use the SVD instead.

Thanks a lot for all your responses. Tim, can you please
explain how to use SVD for finding the determinant?

Basically, I have to solve the system

        det(A(x,c)) = 0

for many different values of scalar c. A can be quite large
(say, 50000 x 50000), but will be sparse with a banded,
albeit quite complex, structure.

Regards,
Deepak


"Tim Davis" <davis@cise.ufl.edu> wrote in message
<g3gdk2$kga$1@fred.mathworks.com>...
> "Deepak " <deepak.trivedi@gmail.com> wrote in message
> <g3ei6q$6vo$1@fred.mathworks.com>...
> > Is it any easier to find the determinant of a banded matrix
> > than a full matrix? Any suggestions what's the best way to
> > do it?
>
> The more basic question is why? Why do you want to compute
> the determinant? If you're worried about performance, then
> that means you're trying to do this for large matrices. If
> that's the case, then you really shouldn't use det at all.
>
> Det is useless for practical computations. It's only of
> theoretical importance. Use the SVD instead.

"Deepak " <deepak.trivedi@gmail.com> wrote in message
<g4bgma$696$1@fred.mathworks.com>...
> Thanks a lot for all your responses. Tim, can you please
> explain how to use SVD for finding the determinant?
>
> Basically, I have to solve the system
>
> det(A(x,c)) = 0
>
> for many different values of scalar c. A can be quite large
> (say, 50000 x 50000), but will be sparse with a banded,
> albeit quite complex, structure.
>
> Regards,
> Deepak

c is a column of A? What is x? "A(x,c)" has no meaning; I
assume it's some kind of expression that's a function of x
and c that gives a matrix A? x is an unknown you're trying
to solve for? Without details it's impossible to suggest
anything.

The SVD won't work on a matrix that large. You should also
avoid the "det" function itself; for such large matrices,
"det" will almost always underflow or overflow.

"Solving" for det(A)=0 sounds like you're trying to find
some kind of null space of the matrix A. In that case,
again, "det" is a horrible way to formulate the problem.

Thanks Tim. I'm really sorry for being so unclear in my
description of the problem. Each entry of A is a function of
x and c. c is a known constant scalar, and x is the unknown
scalar we are solving for.

I am sure I am not formulating the problem in the best
possible way, and I would really appreciate it, if you can
take a look at the equation I am trying to solve, and give
your suggestion regarding the best approach to solve it.

I am trying to solve Eqs. 29, 29a, 29b and 36 of this paper:

Dynamic stability of elastic structures: a finite element
approach

L. Briseghella, C. E. Majorana, and C. Pellegrino

Computers & Structures
Volume 69, Issue 1, October 1998, Pages 11-25

(http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V28-3TVPGF9-2&_user=852002&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000045923&_version=1&_urlVersion=0&_userid=852002&md5=6a35ee465cd2831b94b31127bd0fd602)


In my case Ps is 0, and I have to find pairs (Pd,theta) that
make the determinants in the equations zero.

If you have any problem accessing the paper, please let me know.

Thanks a lot!
Deepak

"Tim Davis" <davis@cise.ufl.edu> wrote in message
<g4l16m$19t$1@fred.mathworks.com>...
> "Deepak " <deepak.trivedi@gmail.com> wrote in message
> <g4bgma$696$1@fred.mathworks.com>...
> > Thanks a lot for all your responses. Tim, can you please
> > explain how to use SVD for finding the determinant?
> >
> > Basically, I have to solve the system
> >
> > det(A(x,c)) = 0
> >
> > for many different values of scalar c. A can be quite large
> > (say, 50000 x 50000), but will be sparse with a banded,
> > albeit quite complex, structure.
> >
> > Regards,
> > Deepak
>
> c is a column of A? What is x? "A(x,c)" has no meaning; I
> assume it's some kind of expression that's a function of x
> and c that gives a matrix A? x is an unknown you're trying
> to solve for? Without details it's impossible to suggest
> anything.
>
> The SVD won't work on a matrix that large. You should also
> avoid the "det" function itself; for such large matrices,
> "det" will almost always underflow or overflow.
>
> "Solving" for det(A)=0 sounds like you're trying to find
> some kind of null space of the matrix A. In that case,
> again, "det" is a horrible way to formulate the problem.

Thanks Tim. I'm really sorry for being so unclear in my
description of the problem. Each entry of A is a function of
x and c. c is a known constant scalar, and x is the unknown
scalar we are solving for.

I am sure I am not formulating the problem in the best
possible way, and I would really appreciate it, if you can
take a look at the equation I am trying to solve, and give
your suggestion regarding the best approach to solve it.

I am trying to solve Eqs. 29, 29a, 29b and 36 of this paper:

Dynamic stability of elastic structures: a finite element
approach

L. Briseghella, C. E. Majorana, and C. Pellegrino

Computers & Structures
Volume 69, Issue 1, October 1998, Pages 11-25

(http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V28-3TVPGF9-2&_user=852002&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000045923&_version=1&_urlVersion=0&_userid=852002&md5=6a35ee465cd2831b94b31127bd0fd602)


In my case Ps is 0, and I have to find pairs (Pd,theta) that
make the determinants in the equations zero.

If you have any problem accessing the paper, please let me know.

Thanks a lot!
Deepak

"Tim Davis" <davis@cise.ufl.edu> wrote in message
<g4l16m$19t$1@fred.mathworks.com>...
> "Deepak " <deepak.trivedi@gmail.com> wrote in message
> <g4bgma$696$1@fred.mathworks.com>...
> > Thanks a lot for all your responses. Tim, can you please
> > explain how to use SVD for finding the determinant?
> >
> > Basically, I have to solve the system
> >
> > det(A(x,c)) = 0
> >
> > for many different values of scalar c. A can be quite large
> > (say, 50000 x 50000), but will be sparse with a banded,
> > albeit quite complex, structure.
> >
> > Regards,
> > Deepak
>
> c is a column of A? What is x? "A(x,c)" has no meaning; I
> assume it's some kind of expression that's a function of x
> and c that gives a matrix A? x is an unknown you're trying
> to solve for? Without details it's impossible to suggest
> anything.
>
> The SVD won't work on a matrix that large. You should also
> avoid the "det" function itself; for such large matrices,
> "det" will almost always underflow or overflow.
>
> "Solving" for det(A)=0 sounds like you're trying to find
> some kind of null space of the matrix A. In that case,
> again, "det" is a horrible way to formulate the problem.

Thanks Tim. I'm really sorry for being so unclear in my
description of the problem. Each entry of A is a function of
x and c. c is a known constant scalar, and x is the unknown
scalar we are solving for.

I am sure I am not formulating the problem in the best
possible way, and I would really appreciate it, if you can
take a look at the equation I am trying to solve, and give
your suggestion regarding the best approach to solve it.

I am trying to solve Eqs. 29, 29a, 29b and 36 of this paper:

Dynamic stability of elastic structures: a finite element
approach

L. Briseghella, C. E. Majorana, and C. Pellegrino

Computers & Structures
Volume 69, Issue 1, October 1998, Pages 11-25

(http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V28-3TVPGF9-2&_user=852002&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000045923&_version=1&_urlVersion=0&_userid=852002&md5=6a35ee465cd2831b94b31127bd0fd602)


In my case Ps is 0, and I have to find pairs (Pd,theta) that
make the determinants in the equations zero.

If you have any problem accessing the paper, please let me know.

Thanks a lot!
Deepak

"Tim Davis" <davis@cise.ufl.edu> wrote in message
<g4l16m$19t$1@fred.mathworks.com>...
> "Deepak " <deepak.trivedi@gmail.com> wrote in message
> <g4bgma$696$1@fred.mathworks.com>...
> > Thanks a lot for all your responses. Tim, can you please
> > explain how to use SVD for finding the determinant?
> >
> > Basically, I have to solve the system
> >
> > det(A(x,c)) = 0
> >
> > for many different values of scalar c. A can be quite large
> > (say, 50000 x 50000), but will be sparse with a banded,
> > albeit quite complex, structure.
> >
> > Regards,
> > Deepak
>
> c is a column of A? What is x? "A(x,c)" has no meaning; I
> assume it's some kind of expression that's a function of x
> and c that gives a matrix A? x is an unknown you're trying
> to solve for? Without details it's impossible to suggest
> anything.
>
> The SVD won't work on a matrix that large. You should also
> avoid the "det" function itself; for such large matrices,
> "det" will almost always underflow or overflow.
>
> "Solving" for det(A)=0 sounds like you're trying to find
> some kind of null space of the matrix A. In that case,
> again, "det" is a horrible way to formulate the problem.

I am trying to solve Eqs. 29, 29a, 29b and 36 of this paper:

Dynamic stability of elastic structures: a finite element
approach

L. Briseghella, C. E. Majorana, and C. Pellegrino

Computers & Structures
Volume 69, Issue 1, October 1998, Pages 11-25

(http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V28-3TVPGF9-2&_user=852002&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000045923&_version=1&_urlVersion=0&_userid=852002&md5=6a35ee465cd2831b94b31127bd0fd602)


In my case Ps is 0, and I have to find pairs (Pd,theta) that
make the determinants in the equations zero.

If you have any problem accessing the paper, please let me know.

Thanks a lot!
Deepak


"Tim Davis" <davis@cise.ufl.edu> wrote in message
<g4l16m$19t$1@fred.mathworks.com>...
> "Deepak " <deepak.trivedi@gmail.com> wrote in message
> <g4bgma$696$1@fred.mathworks.com>...
> > Thanks a lot for all your responses. Tim, can you please
> > explain how to use SVD for finding the determinant?
> >
> > Basically, I have to solve the system
> >
> > det(A(x,c)) = 0
> >
> > for many different values of scalar c. A can be quite large
> > (say, 50000 x 50000), but will be sparse with a banded,
> > albeit quite complex, structure.
> >
> > Regards,
> > Deepak
>
> c is a column of A? What is x? "A(x,c)" has no meaning; I
> assume it's some kind of expression that's a function of x
> and c that gives a matrix A? x is an unknown you're trying
> to solve for? Without details it's impossible to suggest
> anything.
>
> The SVD won't work on a matrix that large. You should also
> avoid the "det" function itself; for such large matrices,
> "det" will almost always underflow or overflow.
>
> "Solving" for det(A)=0 sounds like you're trying to find
> some kind of null space of the matrix A. In that case,
> again, "det" is a horrible way to formulate the problem.

I am trying to solve Eqs. 29, 29a, 29b and 36 of this paper:

Dynamic stability of elastic structures: a finite element
approach

L. Briseghella, C. E. Majorana, and C. Pellegrino

Computers & Structures
Volume 69, Issue 1, October 1998, Pages 11-25

(http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V28-3TVPGF9-2&_user=852002&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000045923&_version=1&_urlVersion=0&_userid=852002&md5=6a35ee465cd2831b94b31127bd0fd602)


In my case Ps is 0, and I have to find pairs (Pd,theta) that
make the determinants in the equations zero.

If you have any problem accessing the paper, please let me know.

Thanks a lot!
Deepak


"Tim Davis" <davis@cise.ufl.edu> wrote in message
<g4l16m$19t$1@fred.mathworks.com>...
> "Deepak " <deepak.trivedi@gmail.com> wrote in message
> <g4bgma$696$1@fred.mathworks.com>...
> > Thanks a lot for all your responses. Tim, can you please
> > explain how to use SVD for finding the determinant?
> >
> > Basically, I have to solve the system
> >
> > det(A(x,c)) = 0
> >
> > for many different values of scalar c. A can be quite large
> > (say, 50000 x 50000), but will be sparse with a banded,
> > albeit quite complex, structure.
> >
> > Regards,
> > Deepak
>
> c is a column of A? What is x? "A(x,c)" has no meaning; I
> assume it's some kind of expression that's a function of x
> and c that gives a matrix A? x is an unknown you're trying
> to solve for? Without details it's impossible to suggest
> anything.
>
> The SVD won't work on a matrix that large. You should also
> avoid the "det" function itself; for such large matrices,
> "det" will almost always underflow or overflow.
>
> "Solving" for det(A)=0 sounds like you're trying to find
> some kind of null space of the matrix A. In that case,
> again, "det" is a horrible way to formulate the problem.

I am trying to solve Eqs. 29, 29a, 29b and 36 of this paper:

Dynamic stability of elastic structures: a finite element
approach

L. Briseghella, C. E. Majorana, and C. Pellegrino

Computers & Structures
Volume 69, Issue 1, October 1998, Pages 11-25

(http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V28-3TVPGF9-2&_user=852002&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000045923&_version=1&_urlVersion=0&_userid=852002&md5=6a35ee465cd2831b94b31127bd0fd602)


In my case Ps is 0, and I have to find pairs (Pd,theta) that
make the determinants in the equations zero.

If you have any problem accessing the paper, please let me know.

Thanks a lot!
Deepak


"Tim Davis" <davis@cise.ufl.edu> wrote in message
<g4l16m$19t$1@fred.mathworks.com>...
> "Deepak " <deepak.trivedi@gmail.com> wrote in message
> <g4bgma$696$1@fred.mathworks.com>...
> > Thanks a lot for all your responses. Tim, can you please
> > explain how to use SVD for finding the determinant?
> >
> > Basically, I have to solve the system
> >
> > det(A(x,c)) = 0
> >
> > for many different values of scalar c. A can be quite large
> > (say, 50000 x 50000), but will be sparse with a banded,
> > albeit quite complex, structure.
> >
> > Regards,
> > Deepak
>
> c is a column of A? What is x? "A(x,c)" has no meaning; I
> assume it's some kind of expression that's a function of x
> and c that gives a matrix A? x is an unknown you're trying
> to solve for? Without details it's impossible to suggest
> anything.
>
> The SVD won't work on a matrix that large. You should also
> avoid the "det" function itself; for such large matrices,
> "det" will almost always underflow or overflow.
>
> "Solving" for det(A)=0 sounds like you're trying to find
> some kind of null space of the matrix A. In that case,
> again, "det" is a horrible way to formulate the problem.

I am trying to solve Eqs. 29, 29a, 29b and 36 of this paper:

Dynamic stability of elastic structures: a finite element
approach

L. Briseghella, C. E. Majorana, and C. Pellegrino

Computers & Structures
Volume 69, Issue 1, October 1998, Pages 11-25

(http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V28-3TVPGF9-2&_user=852002&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000045923&_version=1&_urlVersion=0&_userid=852002&md5=6a35ee465cd2831b94b31127bd0fd602)


In my case Ps is 0, and I have to find pairs (Pd,theta) that
make the determinants in the equations zero.

If you have any problem accessing the paper, please let me know.

Thanks a lot!
Deepak


"Tim Davis" <davis@cise.ufl.edu> wrote in message
<g4l16m$19t$1@fred.mathworks.com>...
> "Deepak " <deepak.trivedi@gmail.com> wrote in message
> <g4bgma$696$1@fred.mathworks.com>...
> > Thanks a lot for all your responses. Tim, can you please
> > explain how to use SVD for finding the determinant?
> >
> > Basically, I have to solve the system
> >
> > det(A(x,c)) = 0
> >
> > for many different values of scalar c. A can be quite large
> > (say, 50000 x 50000), but will be sparse with a banded,
> > albeit quite complex, structure.
> >
> > Regards,
> > Deepak
>
> c is a column of A? What is x? "A(x,c)" has no meaning; I
> assume it's some kind of expression that's a function of x
> and c that gives a matrix A? x is an unknown you're trying
> to solve for? Without details it's impossible to suggest
> anything.
>
> The SVD won't work on a matrix that large. You should also
> avoid the "det" function itself; for such large matrices,
> "det" will almost always underflow or overflow.
>
> "Solving" for det(A)=0 sounds like you're trying to find
> some kind of null space of the matrix A. In that case,
> again, "det" is a horrible way to formulate the problem.