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<H1><A NAME="SEC143" HREF="octave_toc.html#TOC143">Linear Algebra</A></H1>

<P>
This chapter documents the linear algebra functions of Octave.
Reference material for many of these functions may be found in
Golub and Van Loan, <CITE>Matrix Computations, 2nd Ed.</CITE>, Johns Hopkins,
1989, and in <CITE>LAPACK Users' Guide</CITE>, SIAM, 1992.

</P>



<H2><A NAME="SEC144" HREF="octave_toc.html#TOC144">Basic Matrix Functions</A></H2>

<P>
<DL>
<DT><U>Loadable Function:</U> <VAR>aa</VAR> = <B>balance</B> <I>(<VAR>a</VAR>, <VAR>opt</VAR>)</I>
<DD><A NAME="IDX696"></A>
<DT><U>Loadable Function:</U> [<VAR>dd</VAR>, <VAR>aa</VAR>] = <B>balance</B> <I>(<VAR>a</VAR>, <VAR>opt</VAR>)</I>
<DD><A NAME="IDX697"></A>
<DT><U>Loadable Function:</U> [<VAR>cc</VAR>, <VAR>dd</VAR>, <VAR>aa</VAR>, <VAR>bb]</VAR> = <B>balance</B> <I>(<VAR>a</VAR>, <VAR>b</VAR>, <VAR>opt</VAR>)</I>
<DD><A NAME="IDX698"></A>

</P>
<P>
<CODE>[dd, aa] = balance (a)</CODE> returns <CODE>aa = dd \ a * dd</CODE>.
<CODE>aa</CODE> is a matrix whose row and column norms are roughly equal in
magnitude, and <CODE>dd</CODE> = <CODE>p * d</CODE>, where <CODE>p</CODE> is a permutation
matrix and <CODE>d</CODE> is a diagonal matrix of powers of two.  This allows
the equilibration to be computed without roundoff.  Results of
eigenvalue calculation are typically improved by balancing first.

</P>
<P>
<CODE>[cc, dd, aa, bb] = balance (a, b)</CODE> returns <CODE>aa = cc*a*dd</CODE> and
<CODE>bb = cc*b*dd)</CODE>, where <CODE>aa</CODE> and <CODE>bb</CODE> have non-zero
elements of approximately the same magnitude and <CODE>cc</CODE> and <CODE>dd</CODE>
are permuted diagonal matrices as in <CODE>dd</CODE> for the algebraic
eigenvalue problem.

</P>
<P>
The eigenvalue balancing option <CODE>opt</CODE> is selected as follows:

</P>
<DL COMPACT>

<DT><CODE>"N"</CODE>, <CODE>"n"</CODE>
<DD>
No balancing; arguments copied, transformation(s) set to identity.

<DT><CODE>"P"</CODE>, <CODE>"p"</CODE>
<DD>
Permute argument(s) to isolate eigenvalues where possible.

<DT><CODE>"S"</CODE>, <CODE>"s"</CODE>
<DD>
Scale to improve accuracy of computed eigenvalues.

<DT><CODE>"B"</CODE>, <CODE>"b"</CODE>
<DD>
Permute and scale, in that order. Rows/columns of a (and b)
that are isolated by permutation are not scaled.  This is the default
behavior.
</DL>

<P>
Algebraic eigenvalue balancing uses standard LAPACK routines.

</P>
<P>
Generalized eigenvalue problem balancing uses Ward's algorithm
(SIAM Journal on Scientific and Statistical Computing, 1981).
</DL>

</P>
<P>
<DL>
<DT><U>:</U>  <B>cond</B> <I>(<VAR>a</VAR>)</I>
<DD><A NAME="IDX699"></A>
Compute the (two-norm) condition number of a matrix. <CODE>cond (a)</CODE> is
defined as <CODE>norm (a) * norm (inv (a))</CODE>, and is computed via a
singular value decomposition.
</DL>

</P>
<P>
<DL>
<DT><U>Loadable Function:</U>  <B>det</B> <I>(<VAR>a</VAR>)</I>
<DD><A NAME="IDX700"></A>
Compute the determinant of <VAR>a</VAR> using LINPACK.
</DL>

</P>
<P>
<DL>
<DT><U>Loadable Function:</U> <VAR>lambda</VAR> = <B>eig</B> <I>(<VAR>a</VAR>)</I>
<DD><A NAME="IDX701"></A>
<DT><U>Loadable Function:</U> [<VAR>v</VAR>, <VAR>lambda</VAR>] = <B>eig</B> <I>(<VAR>a</VAR>)</I>
<DD><A NAME="IDX702"></A>
The eigenvalues (and eigenvectors) of a matrix are computed in a several
step process which begins with a Hessenberg decomposition, followed by a
Schur decomposition, from which the eigenvalues are apparent.  The
eigenvectors, when desired, are computed by further manipulations of the
Schur decomposition.
</DL>

</P>
<P>
<DL>
<DT><U>Loadable Function:</U> <VAR>G</VAR> = <B>givens</B> <I>(<VAR>x</VAR>, <VAR>y</VAR>)</I>
<DD><A NAME="IDX703"></A>
<DT><U>Loadable Function:</U> [<VAR>c</VAR>, <VAR>s</VAR>] = <B>givens</B> <I>(<VAR>x</VAR>, <VAR>y</VAR>)</I>
<DD><A NAME="IDX704"></A>

</P>
<P>
For example,

</P>

<PRE>
givens (1, 1)
     =>   0.70711   0.70711
         -0.70711   0.70711
</PRE>

</DL>

<P>
<DL>
<DT><U>Loadable Function:</U>  <B>inv</B> <I>(<VAR>a</VAR>)</I>
<DD><A NAME="IDX705"></A>
<DT><U>Loadable Function:</U>  <B>inverse</B> <I>(<VAR>a</VAR>)</I>
<DD><A NAME="IDX706"></A>
Compute the inverse of the square matrix <VAR>a</VAR>.
</DL>

</P>
<P>
<DL>
<DT><U>Function File:</U>  <B>norm</B> <I>(<VAR>a</VAR>, <VAR>p</VAR>)</I>
<DD><A NAME="IDX707"></A>
Compute the p-norm of the matrix <VAR>a</VAR>.  If the second argument is
missing, <CODE>p = 2</CODE> is assumed.

</P>
<P>
If <VAR>a</VAR> is a matrix:

</P>
<DL COMPACT>

<DT><VAR>p</VAR> = <CODE>1</CODE>
<DD>
1-norm, the largest column sum of <VAR>a</VAR>.

<DT><VAR>p</VAR> = <CODE>2</CODE>
<DD>
Largest singular value of <VAR>a</VAR>.

<DT><VAR>p</VAR> = <CODE>Inf</CODE>
<DD>
<A NAME="IDX708"></A>
Infinity norm, the largest row sum of <VAR>a</VAR>.

<DT><VAR>p</VAR> = <CODE>"fro"</CODE>
<DD>
<A NAME="IDX709"></A>
Frobenius norm of <VAR>a</VAR>, <CODE>sqrt (sum (diag (<VAR>a</VAR>' * <VAR>a</VAR>)))</CODE>.
</DL>

<P>
If <VAR>a</VAR> is a vector or a scalar:

</P>
<DL COMPACT>

<DT><VAR>p</VAR> = <CODE>Inf</CODE>
<DD>
<CODE>max (abs (<VAR>a</VAR>))</CODE>.

<DT><VAR>p</VAR> = <CODE>-Inf</CODE>
<DD>
<CODE>min (abs (<VAR>a</VAR>))</CODE>.

<DT>other
<DD>
p-norm of <VAR>a</VAR>, <CODE>(sum (abs (<VAR>a</VAR>) .^ <VAR>p</VAR>)) ^ (1/<VAR>p</VAR>)</CODE>.
</DL>
</DL>

<P>
<DL>
<DT><U>Function File:</U>  <B>null</B> <I>(<VAR>a</VAR>, <VAR>tol</VAR>)</I>
<DD><A NAME="IDX710"></A>
Return an orthonormal basis of the null space of <VAR>a</VAR>.

</P>
<P>
The dimension of the null space is taken as the number of singular
values of <VAR>a</VAR> not greater than <VAR>tol</VAR>.  If the argument <VAR>tol</VAR>
is missing, it is computed as

</P>

<PRE>
max (size (<VAR>a</VAR>)) * max (svd (<VAR>a</VAR>)) * eps
</PRE>

</DL>

<P>
<DL>
<DT><U>Function File:</U>  <B>orth</B> <I>(<VAR>a</VAR>, <VAR>tol</VAR>)</I>
<DD><A NAME="IDX711"></A>
Return an orthonormal basis of the range space of <VAR>a</VAR>.

</P>
<P>
The dimension of the range space is taken as the number of singular
values of <VAR>a</VAR> greater than <VAR>tol</VAR>.  If the argument <VAR>tol</VAR> is
missing, it is computed as

</P>

<PRE>
max (size (<VAR>a</VAR>)) * max (svd (<VAR>a</VAR>)) * eps
</PRE>

</DL>

<P>
<DL>
<DT><U>Function File:</U>  <B>pinv</B> <I>(<VAR>x</VAR>, <VAR>tol</VAR>)</I>
<DD><A NAME="IDX712"></A>
Return the pseudoinverse of <VAR>x</VAR>.  Singular values less than
<VAR>tol</VAR> are ignored. 

</P>
<P>
If the second argument is omitted, it is assumed that

</P>

<PRE>
tol = max (size (<VAR>x</VAR>)) * sigma_max (<VAR>x</VAR>) * eps,
</PRE>

<P>
where <CODE>sigma_max (<VAR>x</VAR>)</CODE> is the maximal singular value of <VAR>x</VAR>.
</DL>

</P>
<P>
<DL>
<DT><U>Function File:</U>  <B>rank</B> <I>(<VAR>a</VAR>, <VAR>tol</VAR>)</I>
<DD><A NAME="IDX713"></A>
Compute the rank of <VAR>a</VAR>, using the singular value decomposition.
The rank is taken to be the number  of singular values of <VAR>a</VAR> that
are greater than the specified tolerance <VAR>tol</VAR>.  If the second
argument is omitted, it is taken to be

</P>

<PRE>
tol = max (size (<VAR>a</VAR>)) * sigma (1) * eps;
</PRE>

<P>
where <CODE>eps</CODE> is machine precision and <CODE>sigma</CODE> is the largest
singular value of <VAR>a</VAR>.
</DL>

</P>
<P>
<DL>
<DT><U>Function File:</U>  <B>trace</B> <I>(<VAR>a</VAR>)</I>
<DD><A NAME="IDX714"></A>
Compute the trace of <VAR>a</VAR>, <CODE>sum (diag (<VAR>a</VAR>))</CODE>.
</DL>

</P>


<H2><A NAME="SEC145" HREF="octave_toc.html#TOC145">Matrix Factorizations</A></H2>

<P>
<DL>
<DT><U>Loadable Function:</U>  <B>chol</B> <I>(<VAR>a</VAR>)</I>
<DD><A NAME="IDX715"></A>
<A NAME="IDX716"></A>
Compute the Cholesky factor, <VAR>r</VAR>, of the symmetric positive definite
matrix <VAR>a</VAR>, where
</DL>

</P>
<P>
<DL>
<DT><U>Loadable Function:</U> <VAR>h</VAR> = <B>hess</B> <I>(<VAR>a</VAR>)</I>
<DD><A NAME="IDX717"></A>
<DT><U>Loadable Function:</U> [<VAR>p</VAR>, <VAR>h</VAR>] = <B>hess</B> <I>(<VAR>a</VAR>)</I>
<DD><A NAME="IDX718"></A>
<A NAME="IDX719"></A>
Compute the Hessenberg decomposition of the matrix <VAR>a</VAR>.

</P>
<P>
The Hessenberg decomposition is usually used as the first step in an
eigenvalue computation, but has other applications as well (see Golub,
Nash, and Van Loan, IEEE Transactions on Automatic Control, 1979.  The
Hessenberg decomposition is
</DL>

</P>
<P>
<DL>
<DT><U>Loadable Function:</U> [<VAR>l</VAR>, <VAR>u</VAR>, <VAR>p</VAR>] = <B>lu</B> <I>(<VAR>a</VAR>)</I>
<DD><A NAME="IDX720"></A>
<A NAME="IDX721"></A>
Compute the LU decomposition of <VAR>a</VAR>, using subroutines from
LAPACK.  The result is returned in a permuted form, according to
the optional return value <VAR>p</VAR>.  For example, given the matrix
<CODE>a = [1, 2; 3, 4]</CODE>,

</P>

<PRE>
[l, u, p] = lu (a)
</PRE>

<P>
returns

</P>

<PRE>
l =

  1.00000  0.00000
  0.33333  1.00000

u =

  3.00000  4.00000
  0.00000  0.66667

p =

  0  1
  1  0
</PRE>

</DL>

<P>
<DL>
<DT><U>Loadable Function:</U> [<VAR>q</VAR>, <VAR>r</VAR>, <VAR>p</VAR>] = <B>qr</B> <I>(<VAR>a</VAR>)</I>
<DD><A NAME="IDX722"></A>
<A NAME="IDX723"></A>
Compute the QR factorization of <VAR>a</VAR>, using standard LAPACK
subroutines.  For example, given the matrix <CODE>a = [1, 2; 3, 4]</CODE>,

</P>

<PRE>
[q, r] = qr (a)
</PRE>

<P>
returns

</P>

<PRE>
q =

  -0.31623  -0.94868
  -0.94868   0.31623

r =

  -3.16228  -4.42719
   0.00000  -0.63246
</PRE>

<P>
The <CODE>qr</CODE> factorization has applications in the solution of least
squares problems
for overdetermined systems of equations (i.e.,
 is a tall, thin matrix).  The QR factorization is

</P>
<P>
The permuted QR factorization <CODE>[<VAR>q</VAR>, <VAR>r</VAR>, <VAR>p</VAR>] =
qr (<VAR>a</VAR>)</CODE> forms the QR factorization such that the diagonal
entries of <CODE>r</CODE> are decreasing in magnitude order.  For example,
given the matrix <CODE>a = [1, 2; 3, 4]</CODE>,

</P>

<PRE>
[q, r, pi] = qr(a)
</PRE>

<P>
returns

</P>

<PRE>
q = 

  -0.44721  -0.89443
  -0.89443   0.44721

r =

  -4.47214  -3.13050
   0.00000   0.44721

p =

   0  1
   1  0
</PRE>

<P>
The permuted <CODE>qr</CODE> factorization <CODE>[q, r, p] = qr (a)</CODE>
factorization allows the construction of an orthogonal basis of
<CODE>span (a)</CODE>.
</DL>

</P>
<P>
<DL>
<DT><U>Loadable Function:</U> <VAR>s</VAR> = <B>schur</B> <I>(<VAR>a</VAR>)</I>
<DD><A NAME="IDX724"></A>
<DT><U>Loadable Function:</U> [<VAR>u</VAR>, <VAR>s</VAR>] = <B>schur</B> <I>(<VAR>a</VAR>, <VAR>opt</VAR>)</I>
<DD><A NAME="IDX725"></A>
<A NAME="IDX726"></A>
The Schur decomposition is used to compute eigenvalues of a
square matrix, and has applications in the solution of algebraic
Riccati equations in control (see <CODE>are</CODE> and <CODE>dare</CODE>).
<CODE>schur</CODE> always returns
where
 is a unitary matrix
and
is upper triangular.  The eigenvalues of
are the diagonal elements of
If the matrix
is real, then the real Schur decomposition is computed, in which the
matrix
is orthogonal and
is block upper triangular
with blocks of size at most
blocks along the diagonal.  The diagonal elements of
(or the eigenvalues of the
blocks, when
appropriate) are the eigenvalues of
and

</P>
<P>
The eigenvalues are optionally ordered along the diagonal according to
the value of <CODE>opt</CODE>.  <CODE>opt = "a"</CODE> indicates that all
eigenvalues with negative real parts should be moved to the leading
block of
(used in <CODE>are</CODE>), <CODE>opt = "d"</CODE> indicates that all eigenvalues
with magnitude less than one should be moved to the leading block of
(used in <CODE>dare</CODE>), and <CODE>opt = "u"</CODE>, the default, indicates that
no ordering of eigenvalues should occur.  The leading
columns of
always span the
subspace corresponding to the
leading eigenvalues of
</DL>

</P>
<P>
<DL>
<DT><U>Loadable Function:</U> <VAR>s</VAR> = <B>svd</B> <I>(<VAR>a</VAR>)</I>
<DD><A NAME="IDX727"></A>
<DT><U>Loadable Function:</U> [<VAR>u</VAR>, <VAR>s</VAR>, <VAR>v</VAR>] = <B>svd</B> <I>(<VAR>a</VAR>)</I>
<DD><A NAME="IDX728"></A>
<A NAME="IDX729"></A>
Compute the singular value decomposition of <VAR>a</VAR>

</P>
<P>
The function <CODE>svd</CODE> normally returns the vector of singular values.
If asked for three return values, it computes
For example,

</P>

<PRE>
svd (hilb (3))
</PRE>

<P>
returns

</P>

<PRE>
ans =

  1.4083189
  0.1223271
  0.0026873
</PRE>

<P>
and

</P>

<PRE>
[u, s, v] = svd (hilb (3))
</PRE>

<P>
returns

</P>

<PRE>
u =

  -0.82704   0.54745   0.12766
  -0.45986  -0.52829  -0.71375
  -0.32330  -0.64901   0.68867

s =

  1.40832  0.00000  0.00000
  0.00000  0.12233  0.00000
  0.00000  0.00000  0.00269

v =

  -0.82704   0.54745   0.12766
  -0.45986  -0.52829  -0.71375
  -0.32330  -0.64901   0.68867
</PRE>

<P>
If given a second argument, <CODE>svd</CODE> returns an economy-sized
decomposition, eliminating the unnecessary rows or columns of <VAR>u</VAR> or
<VAR>v</VAR>.
</DL>

</P>


<H2><A NAME="SEC146" HREF="octave_toc.html#TOC146">Functions of a Matrix</A></H2>

<P>
<DL>
<DT><U>Loadable Function:</U>  <B>expm</B> <I>(<VAR>a</VAR>)</I>
<DD><A NAME="IDX730"></A>
Return the exponential of a matrix, defined as the infinite Taylor
series
The Taylor series is <EM>not</EM> the way to compute the matrix
exponential; see Moler and Van Loan, <CITE>Nineteen Dubious Ways to
Compute the Exponential of a Matrix</CITE>, SIAM Review, 1978.  This routine
uses Ward's diagonal
approximation method with three step preconditioning (SIAM Journal on
Numerical Analysis, 1977).  Diagonal
 approximations are rational polynomials of matrices
 whose Taylor series matches the first
terms of the Taylor series above; direct evaluation of the Taylor series
(with the same preconditioning steps) may be desirable in lieu of the
approximation when
is ill-conditioned.
</DL>

</P>
<P>
<DL>
<DT><U>Loadable Function:</U>  <B>logm</B> <I>(<VAR>a</VAR>)</I>
<DD><A NAME="IDX731"></A>
Compute the matrix logarithm of the square matrix <VAR>a</VAR>.  Note that
this is currently implemented in terms of an eigenvalue expansion and
needs to be improved to be more robust.
</DL>

</P>
<P>
<DL>
<DT><U>Loadable Function:</U>  <B>sqrtm</B> <I>(<VAR>a</VAR>)</I>
<DD><A NAME="IDX732"></A>
Compute the matrix square root of the square matrix <VAR>a</VAR>.  Note that
this is currently implemented in terms of an eigenvalue expansion and
needs to be improved to be more robust.
</DL>

</P>
<P>
<DL>
<DT><U>Function File:</U>  <B>kron</B> <I>(<VAR>a</VAR>, <VAR>b</VAR>)</I>
<DD><A NAME="IDX733"></A>
Form the kronecker product of two matrices, defined block by block as

</P>

<PRE>
x = [a(i, j) b]
</PRE>

<P>
For example,

</P>

<PRE>
kron (1:4, ones (3, 1))
     =>  1  2  3  4
         1  2  3  4
         1  2  3  4
</PRE>

</DL>

<P>
<DL>
<DT><U>Function File:</U> [<VAR>aa</VAR>, <VAR>bb</VAR>, <VAR>q</VAR>, <VAR>z</VAR>] = <B>qzhess</B> <I>(<VAR>a</VAR>, <VAR>b</VAR>)</I>
<DD><A NAME="IDX734"></A>
Compute the Hessenberg-triangular decomposition of the matrix pencil
<CODE>(<VAR>a</VAR>, <VAR>b</VAR>)</CODE>, returning
<CODE><VAR>aa</VAR> = <VAR>q</VAR> * <VAR>a</VAR> * <VAR>z</VAR></CODE>, 
<CODE><VAR>bb</VAR> = <VAR>q</VAR> * <VAR>b</VAR> * <VAR>z</VAR></CODE>, with <VAR>q</VAR> and <VAR>z</VAR>
orthogonal.  For example,

</P>

<PRE>
[aa, bb, q, z] = qzhess ([1, 2; 3, 4], [5, 6; 7, 8])
     => aa = [ -3.02244, -4.41741;  0.92998,  0.69749 ]
     => bb = [ -8.60233, -9.99730;  0.00000, -0.23250 ]
     =>  q = [ -0.58124, -0.81373; -0.81373,  0.58124 ]
     =>  z = [ 1, 0; 0, 1 ]
</PRE>

<P>
The Hessenberg-triangular decomposition is the first step in
Moler and Stewart's QZ decomposition algorithm.

</P>
<P>
Algorithm taken from Golub and Van Loan, <CITE>Matrix Computations, 2nd
edition</CITE>.
</DL>

</P>
<P>
<DL>
<DT><U>Loadable Function:</U>  <B>qzval</B> <I>(<VAR>a</VAR>, <VAR>b</VAR>)</I>
<DD><A NAME="IDX735"></A>
Compute generalized eigenvalues of the matrix pencil 

</P>
<P>
The arguments <VAR>a</VAR> and <VAR>b</VAR> must be real matrices.
</DL>

</P>
<P>
<DL>
<DT><U>Loadable Function:</U> <VAR>x</VAR> = <B>syl</B> <I>(<VAR>a</VAR>, <VAR>b</VAR>, <VAR>c</VAR>)</I>
<DD><A NAME="IDX736"></A>
Solve the Sylvester equation
using standard LAPACK subroutines.  For example,

</P>

<PRE>
syl ([1, 2; 3, 4], [5, 6; 7, 8], [9, 10; 11, 12])
     => [ -0.50000, -0.66667; -0.66667, -0.50000 ]
</PRE>

</DL>

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