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<h1 class="chapter"> 61. linearalgebra </h1>
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<tr><td align="left" valign="top"><a href="#SEC258">61.1 Introduction to linearalgebra</a></td><td> </td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC259">61.2 Functions and Variables for linearalgebra</a></td><td> </td><td align="left" valign="top">
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<p><a name="Item_003a-Introduction-to-linearalgebra"></a>
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<h2 class="section"> 61.1 Introduction to linearalgebra </h2>
<p><code>linearalgebra</code> is a collection of functions for linear algebra.
</p>
<p>Example:
</p>
<pre class="example">(%i1) M : matrix ([1, 2], [1, 2]);
[ 1 2 ]
(%o1) [ ]
[ 1 2 ]
(%i2) nullspace (M);
[ 1 ]
[ ]
(%o2) span([ 1 ])
[ - - ]
[ 2 ]
(%i3) columnspace (M);
[ 1 ]
(%o3) span([ ])
[ 1 ]
(%i4) ptriangularize (M - z*ident(2), z);
[ 1 2 - z ]
(%o4) [ ]
[ 2 ]
[ 0 3 z - z ]
(%i5) M : matrix ([1, 2, 3], [4, 5, 6], [7, 8, 9]) - z*ident(3);
[ 1 - z 2 3 ]
[ ]
(%o5) [ 4 5 - z 6 ]
[ ]
[ 7 8 9 - z ]
(%i6) MM : ptriangularize (M, z);
[ 4 5 - z 6 ]
[ ]
[ 2 ]
[ 66 z 102 z 132 ]
[ 0 -- - -- + ----- + --- ]
(%o6) [ 49 7 49 49 ]
[ ]
[ 3 2 ]
[ 49 z 245 z 147 z ]
[ 0 0 ----- - ------ - ----- ]
[ 264 88 44 ]
(%i7) algebraic : true;
(%o7) true
(%i8) tellrat (MM [3, 3]);
3 2
(%o8) [z - 15 z - 18 z]
(%i9) MM : ratsimp (MM);
[ 4 5 - z 6 ]
[ ]
[ 2 ]
(%o9) [ 66 7 z - 102 z - 132 ]
[ 0 -- - ------------------ ]
[ 49 49 ]
[ ]
[ 0 0 0 ]
(%i10) nullspace (MM);
[ 1 ]
[ ]
[ 2 ]
[ z - 14 z - 16 ]
[ -------------- ]
(%o10) span([ 8 ])
[ ]
[ 2 ]
[ z - 18 z - 12 ]
[ - -------------- ]
[ 12 ]
(%i11) M : matrix ([1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12],
[13, 14, 15, 16]);
[ 1 2 3 4 ]
[ ]
[ 5 6 7 8 ]
(%o11) [ ]
[ 9 10 11 12 ]
[ ]
[ 13 14 15 16 ]
(%i12) columnspace (M);
[ 1 ] [ 2 ]
[ ] [ ]
[ 5 ] [ 6 ]
(%o12) span([ ], [ ])
[ 9 ] [ 10 ]
[ ] [ ]
[ 13 ] [ 14 ]
(%i13) apply ('orthogonal_complement, args (nullspace (transpose (M))));
[ 0 ] [ 1 ]
[ ] [ ]
[ 1 ] [ 0 ]
(%o13) span([ ], [ ])
[ 2 ] [ - 1 ]
[ ] [ ]
[ 3 ] [ - 2 ]
</pre>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Linear-algebra">Linear algebra</a>
·
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</p>
</div>
<p><a name="Item_003a-Functions-and-Variables-for-linearalgebra"></a>
</p><hr size="6">
<a name="Functions-and-Variables-for-linearalgebra"></a>
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<h2 class="section"> 61.2 Functions and Variables for linearalgebra </h2>
<p><a name="Item_003a-addmatrices"></a>
</p><dl>
<dt><u>Function:</u> <b>addmatrices</b><i> (<var>f</var>, <var>M_1</var>, ..., <var>M_n</var>)</i>
<a name="IDX2303"></a>
</dt>
<dd><p>Using the function <var>f</var> as the addition function, return the sum of
the matrices <var>M_1</var>, ..., <var>M_n</var>. The function <var>f</var> must accept any number of
arguments (a Maxima nary function).
</p>
<p>Examples:
</p>
<pre class="example">(%i1) m1 : matrix([1,2],[3,4])$
(%i2) m2 : matrix([7,8],[9,10])$
(%i3) addmatrices('max,m1,m2);
(%o3) matrix([7,8],[9,10])
(%i4) addmatrices('max,m1,m2,5*m1);
(%o4) matrix([7,10],[15,20])
</pre>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-blockmatrixp"></a>
</p><dl>
<dt><u>Function:</u> <b>blockmatrixp</b><i> (<var>M</var>)</i>
<a name="IDX2304"></a>
</dt>
<dd><p>Return true if and only if <var>M</var> is a matrix and every entry of
<var>M</var> is a matrix.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
·
<a href="maxima_95.html#Category_003a-Predicate-functions">Predicate functions</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-columnop"></a>
</p><dl>
<dt><u>Function:</u> <b>columnop</b><i> (<var>M</var>, <var>i</var>, <var>j</var>, <var>theta</var>)</i>
<a name="IDX2305"></a>
</dt>
<dd><p>If <var>M</var> is a matrix, return the matrix that results from doing the
column operation <code>C_i <- C_i - <var>theta</var> * C_j</code>. If <var>M</var> doesn't have a row
<var>i</var> or <var>j</var>, signal an error.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-columnswap"></a>
</p><dl>
<dt><u>Function:</u> <b>columnswap</b><i> (<var>M</var>, <var>i</var>, <var>j</var>)</i>
<a name="IDX2306"></a>
</dt>
<dd><p>If <var>M</var> is a matrix, swap columns <var>i</var> and <var>j</var>. If <var>M</var> doesn't have a column
<var>i</var> or <var>j</var>, signal an error.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-columnspace"></a>
</p><dl>
<dt><u>Function:</u> <b>columnspace</b><i> (<var>M</var>)</i>
<a name="IDX2307"></a>
</dt>
<dd><p>If <var>M</var> is a matrix, return <code>span (v_1, ..., v_n)</code>, where the set
<code>{v_1, ..., v_n}</code> is a basis for the column space of <var>M</var>. The span
of the empty set is <code>{0}</code>. Thus, when the column space has only
one member, return <code>span ()</code>.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-copy"></a>
</p><dl>
<dt><u>Function:</u> <b>copy</b><i> (<var>e</var>)</i>
<a name="IDX2308"></a>
</dt>
<dd><p>Return a copy of the Maxima expression <var>e</var>. Although <var>e</var> can be any
Maxima expression, the copy function is the most useful when <var>e</var> is either
a list or a matrix; consider:
</p><pre class="example">(%i1) m : [1,[2,3]]$
(%i2) mm : m$
(%i3) mm[2][1] : x$
(%i4) m;
(%o4) [1,[x,3]]
(%i5) mm;
(%o5) [1,[x,3]]
</pre><p>Let's try the same experiment, but this time let <var>mm</var> be a copy of <var>m</var>
</p><pre class="example">(%i6) m : [1,[2,3]]$
(%i7) mm : copy(m)$
(%i8) mm[2][1] : x$
(%i9) m;
(%o9) [1,[2,3]]
(%i10) mm;
(%o10) [1,[x,3]]
</pre><p>This time, the assignment to <var>mm</var> does not change the value of <var>m</var>.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-cholesky"></a>
</p><dl>
<dt><u>Function:</u> <b>cholesky</b><i> (<var>M</var>)</i>
<a name="IDX2309"></a>
</dt>
<dt><u>Function:</u> <b>cholesky</b><i> (<var>M</var>, <var>field</var>)</i>
<a name="IDX2310"></a>
</dt>
<dd><p>Return the Cholesky factorization of the matrix selfadjoint (or hermitian) matrix
<var>M</var>. The second argument defaults to 'generalring.' For a description of the
possible values for <var>field</var>, see <code>lu_factor</code>.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Matrix-decompositions">Matrix decompositions</a>
·
<a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-ctranspose"></a>
</p><dl>
<dt><u>Function:</u> <b>ctranspose</b><i> (<var>M</var>)</i>
<a name="IDX2311"></a>
</dt>
<dd><p>Return the complex conjugate transpose of the matrix <var>M</var>. The function
<code>ctranspose</code> uses <code>matrix_element_transpose</code> to transpose each matrix element.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-diag_005fmatrix"></a>
</p><dl>
<dt><u>Function:</u> <b>diag_matrix</b><i> (<var>d_1</var>, <var>d_2</var>,...,<var>d_n</var>)</i>
<a name="IDX2312"></a>
</dt>
<dd><p>Return a diagonal matrix with diagonal entries <var>d_1</var>, <var>d_2</var>,...,<var>d_n</var>.
When the diagonal entries are matrices, the zero entries of the returned matrix
are zero matrices of the appropriate size; for example:
</p><pre class="example">(%i1) diag_matrix(diag_matrix(1,2),diag_matrix(3,4));
[ [ 1 0 ] [ 0 0 ] ]
[ [ ] [ ] ]
[ [ 0 2 ] [ 0 0 ] ]
(%o1) [ ]
[ [ 0 0 ] [ 3 0 ] ]
[ [ ] [ ] ]
[ [ 0 0 ] [ 0 4 ] ]
(%i2) diag_matrix(p,q);
[ p 0 ]
(%o2) [ ]
[ 0 q ]
</pre>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-dotproduct"></a>
</p><dl>
<dt><u>Function:</u> <b>dotproduct</b><i> (<var>u</var>, <var>v</var>)</i>
<a name="IDX2313"></a>
</dt>
<dd><p>Return the dotproduct of vectors <var>u</var> and <var>v</var>. This is the same
as <code>conjugate (transpose (<var>u</var>)) . <var>v</var></code>. The arguments <var>u</var> and <var>v</var> must be
column vectors.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-eigens_005fby_005fjacobi"></a>
</p><dl>
<dt><u>Function:</u> <b>eigens_by_jacobi</b><i> (<var>A</var>)</i>
<a name="IDX2314"></a>
</dt>
<dt><u>Function:</u> <b>eigens_by_jacobi</b><i> (<var>A</var>, <var>field_type</var>)</i>
<a name="IDX2315"></a>
</dt>
<dd><p>Computes the eigenvalues and eigenvectors of <var>A</var> by the method of Jacobi rotations.
<var>A</var> must be a symmetric matrix (but it need not be positive definite nor positive semidefinite).
<var>field_type</var> indicates the computational field, either <code>floatfield</code> or <code>bigfloatfield</code>.
If <var>field_type</var> is not specified, it defaults to <code>floatfield</code>.
</p>
<p>The elements of <var>A</var> must be numbers or expressions which evaluate to numbers
via <code>float</code> or <code>bfloat</code> (depending on <var>field_type</var>).
</p>
<p>Examples:
</p>
<pre class="example">(%i1) S: matrix([1/sqrt(2), 1/sqrt(2)],[-1/sqrt(2), 1/sqrt(2)]);
[ 1 1 ]
[ ------- ------- ]
[ sqrt(2) sqrt(2) ]
(%o1) [ ]
[ 1 1 ]
[ - ------- ------- ]
[ sqrt(2) sqrt(2) ]
(%i2) L : matrix ([sqrt(3), 0], [0, sqrt(5)]);
[ sqrt(3) 0 ]
(%o2) [ ]
[ 0 sqrt(5) ]
(%i3) M : S . L . transpose (S);
[ sqrt(5) sqrt(3) sqrt(5) sqrt(3) ]
[ ------- + ------- ------- - ------- ]
[ 2 2 2 2 ]
(%o3) [ ]
[ sqrt(5) sqrt(3) sqrt(5) sqrt(3) ]
[ ------- - ------- ------- + ------- ]
[ 2 2 2 2 ]
(%i4) eigens_by_jacobi (M);
The largest percent change was 0.1454972243679
The largest percent change was 0.0
number of sweeps: 2
number of rotations: 1
(%o4) [[1.732050807568877, 2.23606797749979],
[ 0.70710678118655 0.70710678118655 ]
[ ]]
[ - 0.70710678118655 0.70710678118655 ]
(%i5) float ([[sqrt(3), sqrt(5)], S]);
(%o5) [[1.732050807568877, 2.23606797749979],
[ 0.70710678118655 0.70710678118655 ]
[ ]]
[ - 0.70710678118655 0.70710678118655 ]
(%i6) eigens_by_jacobi (M, bigfloatfield);
The largest percent change was 1.454972243679028b-1
The largest percent change was 0.0b0
number of sweeps: 2
number of rotations: 1
(%o6) [[1.732050807568877b0, 2.23606797749979b0],
[ 7.071067811865475b-1 7.071067811865475b-1 ]
[ ]]
[ - 7.071067811865475b-1 7.071067811865475b-1 ]
</pre>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Matrix-decompositions">Matrix decompositions</a>
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<a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-get_005flu_005ffactors"></a>
</p><dl>
<dt><u>Function:</u> <b>get_lu_factors</b><i> (<var>x</var>) </i>
<a name="IDX2316"></a>
</dt>
<dd><p>When <code><var>x</var> = lu_factor (<var>A</var>)</code>, then <code>get_lu_factors</code> returns a list of the
form <code>[P, L, U]</code>, where <var>P</var> is a permutation matrix, <var>L</var> is lower triangular with
ones on the diagonal, and <var>U</var> is upper triangular, and <code><var>A</var> = <var>P</var> <var>L</var> <var>U</var></code>.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-hankel"></a>
</p><dl>
<dt><u>Function:</u> <b>hankel</b><i> (<var>col</var>)</i>
<a name="IDX2317"></a>
</dt>
<dt><u>Function:</u> <b>hankel</b><i> (<var>col</var>, <var>row</var>)</i>
<a name="IDX2318"></a>
</dt>
<dd><p>Return a Hankel matrix <var>H</var>. The first column of <var>H</var> is <var>col</var>;
except for the first entry, the last row of <var>H</var> is <var>row</var>. The
default for <var>row</var> is the zero vector with the same length as <var>col</var>.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-hessian"></a>
</p><dl>
<dt><u>Function:</u> <b>hessian</b><i> (<var>f</var>, <var>x</var>)</i>
<a name="IDX2319"></a>
</dt>
<dd><p>Returns the Hessian matrix of <var>f</var> with respect to the list of variables <var>x</var>.
The <code>(i, j)</code>-th element of the Hessian matrix is <code>diff(<var>f</var>, <var>x</var>[i], 1, <var>x</var>[j], 1)</code>.
</p>
<p>Examples:
</p>
<pre class="example">(%i1) hessian (x * sin (y), [x, y]);
[ 0 cos(y) ]
(%o1) [ ]
[ cos(y) - x sin(y) ]
(%i2) depends (F, [a, b]);
(%o2) [F(a, b)]
(%i3) hessian (F, [a, b]);
[ 2 2 ]
[ d F d F ]
[ --- ----- ]
[ 2 da db ]
[ da ]
(%o3) [ ]
[ 2 2 ]
[ d F d F ]
[ ----- --- ]
[ da db 2 ]
[ db ]
</pre>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Differential-calculus">Differential calculus</a>
·
<a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-hilbert_005fmatrix"></a>
</p><dl>
<dt><u>Function:</u> <b>hilbert_matrix</b><i> (<var>n</var>)</i>
<a name="IDX2320"></a>
</dt>
<dd><p>Return the <var>n</var> by <var>n</var> Hilbert matrix. When <var>n</var> isn't a positive
integer, signal an error.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-identfor"></a>
</p><dl>
<dt><u>Function:</u> <b>identfor</b><i> (<var>M</var>)</i>
<a name="IDX2321"></a>
</dt>
<dt><u>Function:</u> <b>identfor</b><i> (<var>M</var>, <var>fld</var>)</i>
<a name="IDX2322"></a>
</dt>
<dd><p>Return an identity matrix that has the same shape as the matrix
<var>M</var>. The diagonal entries of the identity matrix are the
multiplicative identity of the field <var>fld</var>; the default for
<var>fld</var> is <var>generalring</var>.
</p>
<p>The first argument <var>M</var> should be a square matrix or a
non-matrix. When <var>M</var> is a matrix, each entry of <var>M</var> can be a
square matrix - thus <var>M</var> can be a blocked Maxima matrix. The
matrix can be blocked to any (finite) depth.
</p>
<p>See also <code>zerofor</code>
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-invert_005fby_005flu"></a>
</p><dl>
<dt><u>Function:</u> <b>invert_by_lu</b><i> (<var>M</var>, <var>(rng generalring)</var>)</i>
<a name="IDX2323"></a>
</dt>
<dd><p>Invert a matrix <var>M</var> by using the LU factorization. The LU factorization
is done using the ring <var>rng</var>.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-jacobian"></a>
</p><dl>
<dt><u>Function:</u> <b>jacobian</b><i> (<var>f</var>, <var>x</var>)</i>
<a name="IDX2324"></a>
</dt>
<dd><p>Returns the Jacobian matrix of the list of functions <var>f</var> with respect to the list of variables <var>x</var>.
The <code>(i, j)</code>-th element of the Jacobian matrix is <code>diff(<var>f</var>[i], <var>x</var>[j])</code>.
</p>
<p>Examples:
</p>
<pre class="example">(%i1) jacobian ([sin (u - v), sin (u * v)], [u, v]);
[ cos(v - u) - cos(v - u) ]
(%o1) [ ]
[ v cos(u v) u cos(u v) ]
(%i2) depends ([F, G], [y, z]);
(%o2) [F(y, z), G(y, z)]
(%i3) jacobian ([F, G], [y, z]);
[ dF dF ]
[ -- -- ]
[ dy dz ]
(%o3) [ ]
[ dG dG ]
[ -- -- ]
[ dy dz ]
</pre>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Differential-calculus">Differential calculus</a>
·
<a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-kronecker_005fproduct"></a>
</p><dl>
<dt><u>Function:</u> <b>kronecker_product</b><i> (<var>A</var>, <var>B</var>)</i>
<a name="IDX2325"></a>
</dt>
<dd><p>Return the Kronecker product of the matrices <var>A</var> and <var>B</var>.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>listp</b><i> (<var>e</var>, <var>p</var>)</i>
<a name="IDX2326"></a>
</dt>
<dt><u>Function:</u> <b>listp</b><i> (<var>e</var>)</i>
<a name="IDX2327"></a>
</dt>
<dd><p>Given an optional argument <var>p</var>, return <code>true</code> if <var>e</var> is
a Maxima list and <var>p</var> evaluates to <code>true</code> for every list element.
When <code>listp</code> is not given the optional argument, return <code>true</code> if <var>e</var> is
a Maxima list. In all other cases, return <code>false</code>.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
·
<a href="maxima_95.html#Category_003a-Predicate-functions">Predicate functions</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-locate_005fmatrix_005fentry"></a>
</p><dl>
<dt><u>Function:</u> <b>locate_matrix_entry</b><i> (<var>M</var>, <var>r_1</var>, <var>c_1</var>, <var>r_2</var>, <var>c_2</var>, <var>f</var>, <var>rel</var>)</i>
<a name="IDX2328"></a>
</dt>
<dd><p>The first argument must be a matrix; the arguments
<var>r_1</var> through <var>c_2</var> determine a sub-matrix of <var>M</var> that consists of
rows <var>r_1</var> through <var>r_2</var> and columns <var>c_1</var> through <var>c_2</var>.
</p>
<p>Find a entry in the sub-matrix <var>M</var> that satisfies some property.
Three cases:
</p>
<p>(1) <code><var>rel</var> = 'bool</code> and <var>f</var> a predicate:
</p>
<p>Scan the sub-matrix from left to right then top to bottom,
and return the index of the first entry that satisfies the
predicate <var>f</var>. If no matrix entry satisfies <var>f</var>, return <code>false</code>.
</p>
<p>(2) <code><var>rel</var> = 'max</code> and <var>f</var> real-valued:
</p>
<p>Scan the sub-matrix looking for an entry that maximizes <var>f</var>.
Return the index of a maximizing entry.
</p>
<p>(3) <code><var>rel</var> = 'min</code> and <var>f</var> real-valued:
</p>
<p>Scan the sub-matrix looking for an entry that minimizes <var>f</var>.
Return the index of a minimizing entry.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-lu_005fbacksub"></a>
</p><dl>
<dt><u>Function:</u> <b>lu_backsub</b><i> (<var>M</var>, <var>b</var>)</i>
<a name="IDX2329"></a>
</dt>
<dd><p>When <code><var>M</var> = lu_factor (<var>A</var>, <var>field</var>)</code>,
then <code>lu_backsub (<var>M</var>, <var>b</var>)</code> solves the linear
system <code><var>A</var> <var>x</var> = <var>b</var></code>.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-lu_005ffactor"></a>
</p><dl>
<dt><u>Function:</u> <b>lu_factor</b><i> (<var>M</var>, <var>field</var>)</i>
<a name="IDX2330"></a>
</dt>
<dd><p>Return a list of the form <code>[<var>LU</var>, <var>perm</var>, <var>fld</var>]</code>,
or <code>[<var>LU</var>, <var>perm</var>, <var>fld</var>, <var>lower-cnd</var> <var>upper-cnd</var>]</code>, where
</p>
<p> (1) The matrix <var>LU</var> contains the factorization of <var>M</var> in a packed form. Packed
form means three things: First, the rows of <var>LU</var> are permuted according to the
list <var>perm</var>. If, for example, <var>perm</var> is the list <code>[3,2,1]</code>, the actual first row
of the <var>LU</var> factorization is the third row of the matrix <var>LU</var>. Second,
the lower triangular factor of m is the lower triangular part of <var>LU</var> with the
diagonal entries replaced by all ones. Third, the upper triangular factor of
<var>M</var> is the upper triangular part of <var>LU</var>.
</p>
<p> (2) When the field is either <code>floatfield</code> or <code>complexfield</code>,
the numbers <var>lower-cnd</var> and <var>upper-cnd</var> are lower and upper bounds for the
infinity norm condition number of <var>M</var>. For all fields, the condition number
might not be estimated; for such fields, <code>lu_factor</code> returns a two item list.
Both the lower and upper bounds can differ from their true values by
arbitrarily large factors. (See also <code>mat_cond</code>.)
</p>
<p> The argument <var>M</var> must be a square matrix.
</p>
<p> The optional argument <var>fld</var> must be a symbol that determines a ring or field. The pre-defined
fields and rings are:
</p>
<p> (a) <code>generalring</code> - the ring of Maxima expressions,
(b) <code>floatfield</code> - the field of floating point numbers of the type double,
(c) <code>complexfield</code> - the field of complex floating point numbers of the
type double,
(d) <code>crering</code> - the ring of Maxima CRE expressions,
(e) <code>rationalfield</code> - the field of rational numbers,
(f) <code>runningerror</code> - track the all floating point rounding errors,
(g) <code>noncommutingring</code> - the ring of Maxima expressions where multiplication is the
non-commutative dot operator.
</p>
<p>When the field is <code>floatfield</code>, <code>complexfield</code>, or
<code>runningerror</code>, the algorithm uses partial pivoting; for all
other fields, rows are switched only when needed to avoid a zero
pivot.
</p>
<p>Floating point addition arithmetic isn't associative, so the meaning
of 'field' differs from the mathematical definition.
</p>
<p>A member of the field <code>runningerror</code> is a two member Maxima list
of the form <code>[x,n]</code>,where <var>x</var> is a floating point number and
<code>n</code> is an integer. The relative difference between the 'true'
value of <code>x</code> and <code>x</code> is approximately bounded by the machine
epsilon times <code>n</code>. The running error bound drops some terms that
of the order the square of the machine epsilon.
</p>
<p>There is no user-interface for defining a new field. A user that is
familiar with Common Lisp should be able to define a new field. To do
this, a user must define functions for the arithmetic operations and
functions for converting from the field representation to Maxima and
back. Additionally, for ordered fields (where partial pivoting will be
used), a user must define functions for the magnitude and for
comparing field members. After that all that remains is to define a
Common Lisp structure <code>mring</code>. The file <code>mring</code> has many
examples.
</p>
<p>To compute the factorization, the first task is to convert each matrix
entry to a member of the indicated field. When conversion isn't
possible, the factorization halts with an error message. Members of
the field needn't be Maxima expressions. Members of the
<code>complexfield</code>, for example, are Common Lisp complex numbers. Thus
after computing the factorization, the matrix entries must be
converted to Maxima expressions.
</p>
<p>See also <code>get_lu_factors</code>.
</p>
<p>Examples:
</p>
<pre class="example">(%i1) w[i,j] := random (1.0) + %i * random (1.0);
(%o1) w := random(1.) + %i random(1.)
i, j
(%i2) showtime : true$
Evaluation took 0.00 seconds (0.00 elapsed)
(%i3) M : genmatrix (w, 100, 100)$
Evaluation took 7.40 seconds (8.23 elapsed)
(%i4) lu_factor (M, complexfield)$
Evaluation took 28.71 seconds (35.00 elapsed)
(%i5) lu_factor (M, generalring)$
Evaluation took 109.24 seconds (152.10 elapsed)
(%i6) showtime : false$
(%i7) M : matrix ([1 - z, 3], [3, 8 - z]);
[ 1 - z 3 ]
(%o7) [ ]
[ 3 8 - z ]
(%i8) lu_factor (M, generalring);
[ 1 - z 3 ]
[ ]
(%o8) [[ 3 9 ], [1, 2], generalring]
[ ----- - z - ----- + 8 ]
[ 1 - z 1 - z ]
(%i9) get_lu_factors (%);
[ 1 0 ] [ 1 - z 3 ]
[ 1 0 ] [ ] [ ]
(%o9) [[ ], [ 3 ], [ 9 ]]
[ 0 1 ] [ ----- 1 ] [ 0 - z - ----- + 8 ]
[ 1 - z ] [ 1 - z ]
(%i10) %[1] . %[2] . %[3];
[ 1 - z 3 ]
(%o10) [ ]
[ 3 8 - z ]
</pre>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Matrix-decompositions">Matrix decompositions</a>
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<a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-mat_005fcond"></a>
</p><dl>
<dt><u>Function:</u> <b>mat_cond</b><i> (<var>M</var>, 1)</i>
<a name="IDX2331"></a>
</dt>
<dt><u>Function:</u> <b>mat_cond</b><i> (<var>M</var>, inf)</i>
<a name="IDX2332"></a>
</dt>
<dd><p>Return the <var>p</var>-norm matrix condition number of the matrix
<var>m</var>. The allowed values for <var>p</var> are 1 and <var>inf</var>. This
function uses the LU factorization to invert the matrix <var>m</var>. Thus
the running time for <code>mat_cond</code> is proportional to the cube of
the matrix size; <code>lu_factor</code> determines lower and upper bounds
for the infinity norm condition number in time proportional to the
square of the matrix size.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-mat_005fnorm"></a>
</p><dl>
<dt><u>Function:</u> <b>mat_norm</b><i> (<var>M</var>, 1)</i>
<a name="IDX2333"></a>
</dt>
<dt><u>Function:</u> <b>mat_norm</b><i> (<var>M</var>, inf)</i>
<a name="IDX2334"></a>
</dt>
<dt><u>Function:</u> <b>mat_norm</b><i> (<var>M</var>, frobenius)</i>
<a name="IDX2335"></a>
</dt>
<dd><p>Return the matrix <var>p</var>-norm of the matrix <var>M</var>. The allowed values for <var>p</var> are
1, <code>inf</code>, and <code>frobenius</code> (the Frobenius matrix norm). The matrix <var>M</var> should be
an unblocked matrix.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>matrixp</b><i> (<var>e</var>, <var>p</var>)</i>
<a name="IDX2336"></a>
</dt>
<dt><u>Function:</u> <b>matrixp</b><i> (<var>e</var>)</i>
<a name="IDX2337"></a>
</dt>
<dd><p>Given an optional argument <var>p</var>, return <code>true</code> if <var>e</var> is
a matrix and <var>p</var> evaluates to <code>true</code> for every matrix element.
When <code>matrixp</code> is not given an optional argument, return <code>true</code>
if <code>e</code> is a matrix. In all other cases, return <code>false</code>.
</p>
<p>See also <code>blockmatrixp</code>
</p>
<div class=categorybox>
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<a href="maxima_95.html#Category_003a-Predicate-functions">Predicate functions</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-matrix_005fsize"></a>
</p><dl>
<dt><u>Function:</u> <b>matrix_size</b><i> (<var>M</var>)</i>
<a name="IDX2338"></a>
</dt>
<dd><p>Return a two member list that gives the number of rows and columns, respectively
of the matrix <var>M</var>.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-mat_005ffullunblocker"></a>
</p><dl>
<dt><u>Function:</u> <b>mat_fullunblocker</b><i> (<var>M</var>)</i>
<a name="IDX2339"></a>
</dt>
<dd><p>If <var>M</var> is a block matrix, unblock the matrix to all levels. If <var>M</var> is a matrix,
return <var>M</var>; otherwise, signal an error.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-mat_005ftrace"></a>
</p><dl>
<dt><u>Function:</u> <b>mat_trace</b><i> (<var>M</var>)</i>
<a name="IDX2340"></a>
</dt>
<dd><p>Return the trace of the matrix <var>M</var>. If <var>M</var> isn't a matrix, return a
noun form. When <var>M</var> is a block matrix, <code>mat_trace(M)</code> returns
the same value as does <code>mat_trace(mat_unblocker(m))</code>.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-mat_005funblocker"></a>
</p><dl>
<dt><u>Function:</u> <b>mat_unblocker</b><i> (<var>M</var>)</i>
<a name="IDX2341"></a>
</dt>
<dd><p>If <var>M</var> is a block matrix, unblock <var>M</var> one level. If <var>M</var> is a matrix,
<code>mat_unblocker (M)</code> returns <var>M</var>; otherwise, signal an error.
</p>
<p>Thus if each entry of <var>M</var> is matrix, <code>mat_unblocker (M)</code> returns an
unblocked matrix, but if each entry of <var>M</var> is a block matrix, <code>mat_unblocker (M)</code>
returns a block matrix with one less level of blocking.
</p>
<p>If you use block matrices, most likely you'll want to set <code>matrix_element_mult</code> to
<code>"."</code> and <code>matrix_element_transpose</code> to <code>'transpose</code>. See also <code>mat_fullunblocker</code>.
</p>
<p>Example:
</p>
<pre class="example">(%i1) A : matrix ([1, 2], [3, 4]);
[ 1 2 ]
(%o1) [ ]
[ 3 4 ]
(%i2) B : matrix ([7, 8], [9, 10]);
[ 7 8 ]
(%o2) [ ]
[ 9 10 ]
(%i3) matrix ([A, B]);
[ [ 1 2 ] [ 7 8 ] ]
(%o3) [ [ ] [ ] ]
[ [ 3 4 ] [ 9 10 ] ]
(%i4) mat_unblocker (%);
[ 1 2 7 8 ]
(%o4) [ ]
[ 3 4 9 10 ]
</pre>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-nonnegintegerp"></a>
</p><dl>
<dt><u>Function:</u> <b>nonnegintegerp</b><i> (<var>n</var>)</i>
<a name="IDX2342"></a>
</dt>
<dd><p>Return <code>true</code> if and only if <code><var>n</var> >= 0</code> and <var>n</var> is an integer.
</p>
<div class=categorybox>
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<a href="maxima_95.html#Category_003a-Predicate-functions">Predicate functions</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-nullspace"></a>
</p><dl>
<dt><u>Function:</u> <b>nullspace</b><i> (<var>M</var>)</i>
<a name="IDX2343"></a>
</dt>
<dd><p>If <var>M</var> is a matrix, return <code>span (v_1, ..., v_n)</code>, where the set <code>{v_1, ..., v_n}</code>
is a basis for the nullspace of <var>M</var>. The span of the empty set is <code>{0}</code>.
Thus, when the nullspace has only one member, return <code>span ()</code>.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-nullity"></a>
</p><dl>
<dt><u>Function:</u> <b>nullity</b><i> (<var>M</var>)</i>
<a name="IDX2344"></a>
</dt>
<dd><p>If <var>M</var> is a matrix, return the dimension of the nullspace of <var>M</var>.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-orthogonal_005fcomplement"></a>
</p><dl>
<dt><u>Function:</u> <b>orthogonal_complement</b><i> (<var>v_1</var>, ..., <var>v_n</var>)</i>
<a name="IDX2345"></a>
</dt>
<dd><p>Return <code>span (u_1, ..., u_m)</code>, where the set <code>{u_1, ..., u_m}</code> is a
basis for the orthogonal complement of the set <code>(v_1, ..., v_n)</code>.
</p>
<p>Each vector <var>v_1</var> through <var>v_n</var> must be a column vector.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-polynomialp"></a>
</p><dl>
<dt><u>Function:</u> <b>polynomialp</b><i> (<var>p</var>, <var>L</var>, <var>coeffp</var>, <var>exponp</var>)</i>
<a name="IDX2346"></a>
</dt>
<dt><u>Function:</u> <b>polynomialp</b><i> (<var>p</var>, <var>L</var>, <var>coeffp</var>)</i>
<a name="IDX2347"></a>
</dt>
<dt><u>Function:</u> <b>polynomialp</b><i> (<var>p</var>, <var>L</var>)</i>
<a name="IDX2348"></a>
</dt>
<dd><p>Return <code>true</code> if <var>p</var> is a polynomial in the variables in the list <var>L</var>,
The predicate <var>coeffp</var> must evaluate to <code>true</code> for each
coefficient, and the predicate <var>exponp</var> must evaluate to <code>true</code> for all
exponents of the variables in <var>L</var>. If you want to use a non-default
value for <var>exponp</var>, you must supply <var>coeffp</var> with a value even if you want
to use the default for <var>coeffp</var>.
</p>
<p><code>polynomialp (<var>p</var>, <var>L</var>, <var>coeffp</var>)</code> is equivalent to
<code>polynomialp (<var>p</var>, <var>L</var>, <var>coeffp</var>, 'nonnegintegerp)</code>.
</p>
<p><code>polynomialp (<var>p</var>, <var>L</var>)</code> is equivalent to
<code>polynomialp (<var>p</var>, L<var>,</var> 'constantp, 'nonnegintegerp)</code>.
</p>
<p>The polynomial needn't be expanded:
</p>
<pre class="example">(%i1) polynomialp ((x + 1)*(x + 2), [x]);
(%o1) true
(%i2) polynomialp ((x + 1)*(x + 2)^a, [x]);
(%o2) false
</pre>
<p>An example using non-default values for coeffp and exponp:
</p>
<pre class="example">(%i1) polynomialp ((x + 1)*(x + 2)^(3/2), [x], numberp, numberp);
(%o1) true
(%i2) polynomialp ((x^(1/2) + 1)*(x + 2)^(3/2), [x], numberp,
numberp);
(%o2) true
</pre>
<p>Polynomials with two variables:
</p>
<pre class="example">(%i1) polynomialp (x^2 + 5*x*y + y^2, [x]);
(%o1) false
(%i2) polynomialp (x^2 + 5*x*y + y^2, [x, y]);
(%o2) true
</pre>
<div class=categorybox>
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<a href="maxima_95.html#Category_003a-Predicate-functions">Predicate functions</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-polytocompanion"></a>
</p><dl>
<dt><u>Function:</u> <b>polytocompanion</b><i> (<var>p</var>, <var>x</var>)</i>
<a name="IDX2349"></a>
</dt>
<dd><p>If <var>p</var> is a polynomial in <var>x</var>, return the companion matrix of <var>p</var>. For
a monic polynomial <var>p</var> of degree <var>n</var>,
we have <code><var>p</var> = (-1)^<var>n</var> charpoly (polytocompanion (<var>p</var>, <var>x</var>))</code>.
</p>
<p>When <var>p</var> isn't a polynomial in <var>x</var>, signal an error.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-ptriangularize"></a>
</p><dl>
<dt><u>Function:</u> <b>ptriangularize</b><i> (<var>M</var>, <var>v</var>)</i>
<a name="IDX2350"></a>
</dt>
<dd><p>If <var>M</var> is a matrix with each entry a polynomial in <var>v</var>, return
a matrix <var>M2</var> such that
</p>
<p>(1) <var>M2</var> is upper triangular,
</p>
<p>(2) <code><var>M2</var> = <var>E_n</var> ... <var>E_1</var> <var>M</var></code>,
where <var>E_1</var> through <var>E_n</var> are elementary matrices
whose entries are polynomials in <var>v</var>,
</p>
<p>(3) <code>|det (<var>M</var>)| = |det (<var>M2</var>)|</code>,
</p>
<p>Note: This function doesn't check that every entry is a polynomial in <var>v</var>.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-rowop"></a>
</p><dl>
<dt><u>Function:</u> <b>rowop</b><i> (<var>M</var>, <var>i</var>, <var>j</var>, <var>theta</var>)</i>
<a name="IDX2351"></a>
</dt>
<dd><p>If <var>M</var> is a matrix, return the matrix that results from doing the
row operation <code>R_i <- R_i - theta * R_j</code>. If <var>M</var> doesn't have a row
<var>i</var> or <var>j</var>, signal an error.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>rank</b><i> (<var>M</var>)</i>
<a name="IDX2352"></a>
</dt>
<dd><p>Return the rank of that matrix <var>M</var>. The rank is the dimension of the
column space. Example:
</p><pre class="example">(%i1) rank(matrix([1,2],[2,4]));
(%o1) 1
(%i2) rank(matrix([1,b],[c,d]));
Proviso: {d - b c # 0}
(%o2) 2
</pre>
<div class=categorybox>
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</p>
</div>
</dd></dl>
<p><a name="Item_003a-rowswap"></a>
</p><dl>
<dt><u>Function:</u> <b>rowswap</b><i> (<var>M</var>, <var>i</var>, <var>j</var>)</i>
<a name="IDX2353"></a>
</dt>
<dd><p>If <var>M</var> is a matrix, swap rows <var>i</var> and <var>j</var>. If <var>M</var> doesn't have a row
<var>i</var> or <var>j</var>, signal an error.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-toeplitz"></a>
</p><dl>
<dt><u>Function:</u> <b>toeplitz</b><i> (<var>col</var>)</i>
<a name="IDX2354"></a>
</dt>
<dt><u>Function:</u> <b>toeplitz</b><i> (<var>col</var>, <var>row</var>)</i>
<a name="IDX2355"></a>
</dt>
<dd><p>Return a Toeplitz matrix <var>T</var>. The first first column of <var>T</var> is <var>col</var>;
except for the first entry, the first row of <var>T</var> is <var>row</var>. The
default for <var>row</var> is complex conjugate of <var>col</var>. Example:
</p><pre class="example">(%i1) toeplitz([1,2,3],[x,y,z]);
[ 1 y z ]
[ ]
(%o1) [ 2 1 y ]
[ ]
[ 3 2 1 ]
(%i2) toeplitz([1,1+%i]);
[ 1 1 - %I ]
(%o2) [ ]
[ %I + 1 1 ]
</pre>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-vandermonde_005fmatrix"></a>
</p><dl>
<dt><u>Function:</u> <b>vandermonde_matrix</b><i> ([<var>x_1</var>, ..., <var>x_n</var>])</i>
<a name="IDX2356"></a>
</dt>
<dd><p>Return a <var>n</var> by <var>n</var> matrix whose <var>i</var>-th row is
<code>[1, <var>x_i</var>, <var>x_i</var>^2, ... <var>x_i</var>^(<var>n</var>-1)]</code>.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-zerofor"></a>
</p><dl>
<dt><u>Function:</u> <b>zerofor</b><i> (<var>M</var>)</i>
<a name="IDX2357"></a>
</dt>
<dt><u>Function:</u> <b>zerofor</b><i> (<var>M</var>, <var>fld</var>)</i>
<a name="IDX2358"></a>
</dt>
<dd><p>Return a zero matrix that has the same shape as the matrix
<var>M</var>. Every entry of the zero matrix is the
additive identity of the field <var>fld</var>; the default for
<var>fld</var> is <var>generalring</var>.
</p>
<p>The first argument <var>M</var> should be a square matrix or a
non-matrix. When <var>M</var> is a matrix, each entry of <var>M</var> can be a
square matrix - thus <var>M</var> can be a blocked Maxima matrix. The
matrix can be blocked to any (finite) depth.
</p>
<p>See also <code>identfor</code>
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-zeromatrixp"></a>
</p><dl>
<dt><u>Function:</u> <b>zeromatrixp</b><i> (<var>M</var>)</i>
<a name="IDX2359"></a>
</dt>
<dd><p>If <var>M</var> is not a block matrix, return <code>true</code> if <code>is (equal (<var>e</var>, 0))</code>
is true for each element <var>e</var> of the matrix <var>M</var>. If <var>M</var> is a block matrix, return
<code>true</code> if <code>zeromatrixp</code> evaluates to <code>true</code> for each element of <var>e</var>.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-linearalgebra">Package linearalgebra</a>
·
<a href="maxima_95.html#Category_003a-Predicate-functions">Predicate functions</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-lsquares"></a>
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