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<div class="refentry" title="Linear Algebra Opcodes">
<a id="linearalgebraopcodes"></a>
<div class="titlepage"></div>
<a id="Indexlinearalgebra" class="indexterm"></a>
<div class="refnamediv">
<h2>
<span class="refentrytitle">Linear Algebra Opcodes</span>
</h2>
<p>Linear Algebra Opcodes —
Scalar, vector, and matrix arithmetic on real and complex values.
</p>
</div>
<div class="refsect1" title="Description">
<a id="idp10373040"></a>
<h2>Description</h2>
<p>These opcodes implement many linear algebra operations,
from scalar, vector, and matrix arithmetic up to
and including QR based eigenvalue decompositions.
The opcodes are designed for digital signal processing,
and of course other mathematical operations,
in the Csound orchestra language.</p>
<p>The numerical implementation uses the gmm++ library
from <a class="ulink" href="http://home.gna.org/getfem/gmm_intro" target="_top"><em class="citetitle">home.gna.org/getfem/gmm_intro</em></a>.</p>
<div class="warning" title="Warning" style="margin-left: 0.5in; margin-right: 0.5in;">
<table border="0" summary="Warning">
<tr>
<td rowspan="2" align="center" valign="top" width="25">
<img alt="[Warning]" src="images/warning.png" />
</td>
<th align="left">Warning</th>
</tr>
<tr>
<td align="left" valign="top">
<p>
For applications with f-sig variables, array arithmetic must be
performed only when the f-sig is "current," because f-rate
is some fraction of k-rate; currency can be determined with the
la_k_current_f opcode.
</p>
<p>
For applications using assignments between real vectors and
a-rate variables, array arithmetic must be performed only when the
vectors are "current", because the size of the vector may be
some integral multiple of ksmps; currency can be determined
by means of the la_k_current_vr opcode.
</p>
</td>
</tr>
</table>
</div>
<div class="table">
<a id="LinearAlgebraDataTypes"></a>
<p class="title">
<strong>Table 5. Linear Algebra Data Types</strong>
</p>
<div class="table-contents">
<table summary="Linear Algebra Data Types" border="1">
<colgroup>
<col />
<col />
<col />
</colgroup>
<thead>
<tr>
<th>Mathematical Type</th>
<th>Code</th>
<th>Corresponding Csound Type or Types</th>
</tr>
</thead>
<tbody>
<tr>
<td>real scalar </td>
<td>r</td>
<td>i-rate or k-rate variable</td>
</tr>
<tr>
<td>complex scalar</td>
<td>c</td>
<td>pair of i-rate or k-rate variables, e.g. "kr, ki"</td>
</tr>
<tr>
<td>real vector</td>
<td>vr</td>
<td>i-rate variable holding address of array</td>
</tr>
<tr>
<td>real vector</td>
<td>a</td>
<td>a-rate variable</td>
</tr>
<tr>
<td>real vector</td>
<td>t</td>
<td>function table number</td>
</tr>
<tr>
<td>complex vector</td>
<td>vc</td>
<td>i-rate variable holding address of array</td>
</tr>
<tr>
<td>complex vector</td>
<td>f</td>
<td>fsig variable</td>
</tr>
<tr>
<td>real matrix</td>
<td>mr</td>
<td>i-rate variable holding address of array</td>
</tr>
<tr>
<td>complex matrix</td>
<td>mc</td>
<td>i-rate variable holding address of array</td>
</tr>
</tbody>
</table>
</div>
</div>
<br class="table-break" />
<p>All arrays are 0-based; the first index iterates rows to give columns,
the second index iterates columns to give elements.</p>
<p>All arrays are general and dense; banded, Hermitian, symmetric
and sparse routines are not implemented.</p>
<p>An array can be of type code vr, vc, mr, or mc
and is stored in an i-rate object.
In orchestra code, an array is passed as a
MYFLT i-rate variable that contains the address
of the array object, which is actually stored
in the allocator opcode instance.
Although array variables are i-rate, of course
their values and even shapes may change at i-rate or k-rate.
</p>
<p>All operands must be pre-allocated; except for the creation
opcodes, no opcode ever allocates any arrays.
This is true even if the array appears on the
left-hand side of an opcode! However, some operations
may reshape arrays to hold results.
</p>
<p>
Arrays are automatically deallocated when their instrument
is deallocated.
</p>
<p>
Not only for more efficient performance,
but also to make it easier to remember opcode names,
the performance rate, output value types, operation names,
and input value types are deterministically
encoded into the opcode name:
</p>
<div class="orderedlist">
<ol class="orderedlist" type="1">
<li class="listitem">"la" for "linear algebra opcode family".</li>
<li class="listitem">"i" or "k" for performance rate.</li>
<li class="listitem">Type code(s) (see above table) for output value(s), but only if the type is not implicit from the input values.</li>
<li class="listitem">Operation name: common mathematical name (preferred) or abbreviation.</li>
<li class="listitem">Type code(s) for input values, if not implicit.</li>
</ol>
</div>
<p>For additional details, see the gmm++ documentation at
<a class="ulink" href="http://download.gna.org/getfem/doc/gmmuser.pdf" target="_top"><em class="citetitle">http://download.gna.org/getfem/doc/gmmuser.pdf</em></a>.
</p>
</div>
<div class="refsect1" title="Syntax">
<a id="idp10410744"></a>
<h2>Syntax</h2>
<h4><a id="idp10411064"></a>
Array Creation
</h4>
<pre class="synopsis">ivr <span class="command"><strong>la_i_vr_create</strong></span> irows</pre>
<p>
Create a real vector with irows rows.
</p>
<pre class="synopsis">ivc <span class="command"><strong>la_i_vc_create</strong></span> irows</pre>
<p>
Create a complex vector with irows rows.
</p>
<pre class="synopsis">imr <span class="command"><strong>la_i_mr_create</strong></span> irows, icolumns [, odiagonal]</pre>
<p>
Create a real matrix with irows rows and icolumns columns, with an optional value on the diagonal.
</p>
<pre class="synopsis">imc <span class="command"><strong>la_i_mc_create</strong></span> irows, icolumns [, odiagonal_r, odiagonal_i]</pre>
<p>
Create a complex matrix with irows rows and icolumns columns, with an optional complex value on the diagonal.
</p>
<h4><a id="idp10415160"></a>
Array Introspection
</h4>
<pre class="synopsis">irows <span class="command"><strong>la_i_size_vr</strong></span> ivr</pre>
<p>Return the number of rows in real vector ivr.</p>
<pre class="synopsis">irows <span class="command"><strong>la_i_size_vc</strong></span> ivc</pre>
<p>Return the number of rows in complex vector ivc.</p>
<pre class="synopsis">irows, icolumns <span class="command"><strong>la_i_size_mr</strong></span> imr</pre>
<p>Return the number of rows and columns in real matrix imr.</p>
<pre class="synopsis">irows, icolumns <span class="command"><strong>la_i_size_mc</strong></span> imc</pre>
<p>Return the number of rows and columns in complex matrix imc.</p>
<pre class="synopsis">kfiscurrent <span class="command"><strong>la_k_current_f</strong></span> fsig</pre>
<p>Return 1 if fsig is current, that is, if the value of fsig will change on the next kperiod.</p>
<pre class="synopsis">kvriscurrent <span class="command"><strong>la_k_current_vr</strong></span> ivr</pre>
<p>Return 1 if the real vector ivr is current, that is, if Csound's current audio sample frame stands at index 0 of the vector.</p>
<pre class="synopsis"> <span class="command"><strong>la_i_print_vr</strong></span> ivr</pre>
<p>Print the value of real vector ivr.</p>
<pre class="synopsis"> <span class="command"><strong>la_i_print_vc</strong></span> ivc</pre>
<p>Print the value of complex vector ivc.</p>
<pre class="synopsis"> <span class="command"><strong>la_i_print_mr</strong></span> imr</pre>
<p>Print the value of real matrix imr.</p>
<pre class="synopsis"> <span class="command"><strong>la_i_print_mc</strong></span> imc</pre>
<p>Print the value of complex matrix imc.</p>
<h4><a id="idp10424448"></a>
Array Assignment and Conversion
</h4>
<pre class="synopsis">ivr <span class="command"><strong>la_i_assign_vr</strong></span> ivr</pre>
<p>Assign the value of the real vector on the right-hand side to the real vector on the left-hand side, at i-rate.</p>
<pre class="synopsis">ivr <span class="command"><strong>la_k_assign_vr</strong></span> ivr</pre>
<p>Assign the value of the real vector on the right-hand side to the real vector on the left-hand side, at k-rate.</p>
<pre class="synopsis">ivc <span class="command"><strong>la_i_assign_vc</strong></span> ivc</pre>
<pre class="synopsis">ivc <span class="command"><strong>la_k_assign_vc</strong></span> ivr</pre>
<pre class="synopsis">imr <span class="command"><strong>la_i_assign_mr</strong></span> imr</pre>
<pre class="synopsis">imr <span class="command"><strong>la_k_assign_mr</strong></span> imr</pre>
<pre class="synopsis">imc <span class="command"><strong>la_i_assign_mc</strong></span> imc</pre>
<pre class="synopsis">imc <span class="command"><strong>la_k_assign_mc</strong></span> imr</pre>
<div class="warning" title="Warning" style="margin-left: 0.5in; margin-right: 0.5in;">
<table border="0" summary="Warning">
<tr>
<td rowspan="2" align="center" valign="top" width="25">
<img alt="[Warning]" src="images/warning.png" />
</td>
<th align="left">Warning</th>
</tr>
<tr>
<td align="left" valign="top">
<p>
Assignments to vectors from tables or fsigs may resize the vectors.
</p>
<p>
Assignments to vectors from a-rate variables, or to a-rate variables
from vectors, will be performed incrementally, one chunk of ksmps elements
per kperiod. Therefore, array arithmetic on such vectors should only be
performed when the vectors are current, as determined by the la_k_currrent_vr opcode.
</p>
</td>
</tr>
</table>
</div>
<pre class="synopsis">ivr <span class="command"><strong>la_k_assign_a</strong></span> asig</pre>
<pre class="synopsis">ivr <span class="command"><strong>la_i_assign_t</strong></span> itablenumber</pre>
<pre class="synopsis">ivr <span class="command"><strong>la_k_assign_t</strong></span> itablenumber</pre>
<pre class="synopsis">ivc <span class="command"><strong>la_k_assign_f</strong></span> fsig</pre>
<pre class="synopsis">asig <span class="command"><strong>la_k_a_assign</strong></span> ivr</pre>
<pre class="synopsis">itablenum <span class="command"><strong>la_i_t_assign</strong></span> ivr</pre>
<pre class="synopsis">itablenum <span class="command"><strong>la_k_t_assign</strong></span> ivr</pre>
<pre class="synopsis">fsig <span class="command"><strong>la_k_f_assign</strong></span> ivc</pre>
<h4><a id="idp10436920"></a>
Fill Arrays with Random Elements
</h4>
<pre class="synopsis">ivr <span class="command"><strong>la_i_random_vr</strong></span> [ifill_fraction]</pre>
<pre class="synopsis">ivr <span class="command"><strong>la_k_random_vr</strong></span> [kfill_fraction]</pre>
<pre class="synopsis">ivc <span class="command"><strong>la_i_random_vc</strong></span> [ifill_fraction]</pre>
<pre class="synopsis">ivc <span class="command"><strong>la_k_random_vc</strong></span> [kfill_fraction]</pre>
<pre class="synopsis">imr <span class="command"><strong>la_i_random_mr</strong></span> [ifill_fraction]</pre>
<pre class="synopsis">imr <span class="command"><strong>la_k_random_mr</strong></span> [kfill_fraction]</pre>
<pre class="synopsis">imc <span class="command"><strong>la_i_random_mc</strong></span> [ifill_fraction]</pre>
<pre class="synopsis">imc <span class="command"><strong>la_k_random_mc</strong></span> [kfill_fraction]</pre>
<h4><a id="idp10442480"></a>
Array Element Access
</h4>
<pre class="synopsis">ivr <span class="command"><strong>la_i_vr_set</strong></span> irow, ivalue</pre>
<pre class="synopsis">kvr <span class="command"><strong>la_k_vr_set</strong></span> krow, kvalue</pre>
<pre class="synopsis">ivc <span class="command"><strong>la_i_vc_set</strong></span> irow, ivalue_r, ivalue_i</pre>
<pre class="synopsis">kvc <span class="command"><strong>la_k_vc_set</strong></span> krow, kvalue_r, kvalue_i</pre>
<pre class="synopsis">imr <span class="command"><strong>la_i mr_set</strong></span> irow, icolumn, ivalue</pre>
<pre class="synopsis">kmr <span class="command"><strong>la_k mr_set</strong></span> krow, kcolumn, ivalue</pre>
<pre class="synopsis">imc <span class="command"><strong>la_i_mc_set</strong></span> irow, icolumn, ivalue_r, ivalue_i</pre>
<pre class="synopsis">kmc <span class="command"><strong>la_k_mc_set</strong></span> krow, kcolumn, kvalue_r, kvalue_i</pre>
<pre class="synopsis">ivalue <span class="command"><strong>la_i_get_vr</strong></span> ivr, irow</pre>
<pre class="synopsis">kvalue <span class="command"><strong>la_k_get_vr</strong></span> ivr, krow</pre>
<pre class="synopsis">ivalue_r, ivalue_i <span class="command"><strong>la_i_get_vc</strong></span> ivc, irow</pre>
<pre class="synopsis">kvalue_r, kvalue_i <span class="command"><strong>la_k_get_vc</strong></span> ivc, krow</pre>
<pre class="synopsis">ivalue <span class="command"><strong>la_i_get_mr</strong></span> imr, irow, icolumn</pre>
<pre class="synopsis">kvalue <span class="command"><strong>la_k_get_mr</strong></span> imr, krow, kcolumn</pre>
<pre class="synopsis">ivalue_r, ivalue_i <span class="command"><strong>la_i_get_mc</strong></span> imc, irow, icolumn</pre>
<pre class="synopsis">kvalue_r, kvalue_i <span class="command"><strong>la_k_get_mc</strong></span> imc, krow, kcolumn</pre>
<h4><a id="idp10453768"></a>
Single Array Operations
</h4>
<pre class="synopsis">imr <span class="command"><strong>la_i_transpose_mr</strong></span> imr</pre>
<pre class="synopsis">imr <span class="command"><strong>la_k_transpose_mr</strong></span> imr</pre>
<pre class="synopsis">imc <span class="command"><strong>la_i_transpose_mc</strong></span> imc</pre>
<pre class="synopsis">imc <span class="command"><strong>la_k_transpose_mc</strong></span> imc</pre>
<pre class="synopsis">ivr <span class="command"><strong>la_i_conjugate_vr</strong></span> ivr</pre>
<pre class="synopsis">ivr <span class="command"><strong>la_k_conjugate_vr</strong></span> ivr</pre>
<pre class="synopsis">ivc <span class="command"><strong>la_i_conjugate_vc</strong></span> ivc</pre>
<pre class="synopsis">ivc <span class="command"><strong>la_k_conjugate_vc</strong></span> ivc</pre>
<pre class="synopsis">imr <span class="command"><strong>la_i_conjugate_mr</strong></span> imr</pre>
<pre class="synopsis">imr <span class="command"><strong>la_k_conjugate_mr</strong></span> imr</pre>
<pre class="synopsis">imc <span class="command"><strong>la_i_conjugate_mc</strong></span> imc</pre>
<pre class="synopsis">imc <span class="command"><strong>la_k_conjugate_mc</strong></span> imc</pre>
<h4><a id="idp10461968"></a>
Scalar Operations
</h4>
<pre class="synopsis">ir <span class="command"><strong>la_i_norm1_vr</strong></span> ivr</pre>
<pre class="synopsis">kr <span class="command"><strong>la_k_norm1_vr</strong></span> ivc</pre>
<pre class="synopsis">ir <span class="command"><strong>la_i_norm1_vc</strong></span> ivc</pre>
<pre class="synopsis">kr <span class="command"><strong>la_k_norm1_vc</strong></span> ivc</pre>
<pre class="synopsis">ir <span class="command"><strong>la_i_norm1_mr</strong></span> imr</pre>
<pre class="synopsis">kr <span class="command"><strong>la_k_norm1_mr</strong></span> imr</pre>
<pre class="synopsis">ir <span class="command"><strong>la_i_norm1_mc</strong></span> imc</pre>
<pre class="synopsis">kr <span class="command"><strong>la_k_norm1_mc</strong></span> imc</pre>
<pre class="synopsis">ir <span class="command"><strong>la_i_norm_euclid_vr</strong></span> ivr</pre>
<pre class="synopsis">kr <span class="command"><strong>la_k_norm_euclid_vr</strong></span> ivr</pre>
<pre class="synopsis">ir <span class="command"><strong>la_i_norm_euclid_vc</strong></span> ivc</pre>
<pre class="synopsis">kr <span class="command"><strong>la_k_norm_euclid_vc</strong></span> ivc</pre>
<pre class="synopsis">ir <span class="command"><strong>la_i_norm_euclid_mr</strong></span> mvr</pre>
<pre class="synopsis">kr <span class="command"><strong>la_k_norm_euclid_mr</strong></span> mvr</pre>
<pre class="synopsis">ir <span class="command"><strong>la_i_norm_euclid_mc</strong></span> mvc</pre>
<pre class="synopsis">kr <span class="command"><strong>la_k_norm_euclid_mc</strong></span> mvc</pre>
<pre class="synopsis">ir <span class="command"><strong>la_i_distance_vr</strong></span> ivr</pre>
<pre class="synopsis">kr <span class="command"><strong>la_k_distance_vr</strong></span> ivr</pre>
<pre class="synopsis">ir <span class="command"><strong>la_i_distance_vc</strong></span> ivc</pre>
<pre class="synopsis">kr <span class="command"><strong>la_k_distance_vc</strong></span> ivc</pre>
<pre class="synopsis">ir <span class="command"><strong>la_i_norm_max</strong></span> imr</pre>
<pre class="synopsis">kr <span class="command"><strong>la_k_norm_max</strong></span> imc</pre>
<pre class="synopsis">ir <span class="command"><strong>la_i_norm_max</strong></span> imr</pre>
<pre class="synopsis">kr <span class="command"><strong>la_k_norm_max</strong></span> imc</pre>
<pre class="synopsis">ir <span class="command"><strong>la_i_norm_inf_vr</strong></span> ivr</pre>
<pre class="synopsis">kr <span class="command"><strong>la_k_norm_inf_vr</strong></span> ivr</pre>
<pre class="synopsis">ir <span class="command"><strong>la_i_norm_inf_vc</strong></span> ivc</pre>
<pre class="synopsis">kr <span class="command"><strong>la_k_norm_inf_vc</strong></span> ivc</pre>
<pre class="synopsis">ir <span class="command"><strong>la_i_norm_inf_mr</strong></span> imr</pre>
<pre class="synopsis">kr <span class="command"><strong>la_k_norm_inf_mr</strong></span> imr</pre>
<pre class="synopsis">ir <span class="command"><strong>la_i_norm_inf_mc</strong></span> imc</pre>
<pre class="synopsis">kr <span class="command"><strong>la_k_norm_inf_mc</strong></span> imc</pre>
<pre class="synopsis">ir <span class="command"><strong>la_i_trace_mr</strong></span> imr</pre>
<pre class="synopsis">kr <span class="command"><strong>la_k_trace_mr</strong></span> imr</pre>
<pre class="synopsis">ir, ii <span class="command"><strong>la_i_trace_mc</strong></span> imc</pre>
<pre class="synopsis">kr, ki <span class="command"><strong>la_k_trace_mc</strong></span> imc</pre>
<pre class="synopsis">ir <span class="command"><strong>la_i_lu_det</strong></span> imr</pre>
<pre class="synopsis">kr <span class="command"><strong>la_k_lu_det</strong></span> imr</pre>
<pre class="synopsis">ir <span class="command"><strong>la_i_lu_det</strong></span> imc</pre>
<pre class="synopsis">kr <span class="command"><strong>la_k_lu_det</strong></span> imc</pre>
<h4><a id="idp10488760"></a>
Elementwise Array-Array Operations
</h4>
<pre class="synopsis">ivr <span class="command"><strong>la_i_add_vr</strong></span> ivr_a, ivr_b</pre>
<pre class="synopsis">ivc <span class="command"><strong>la_k_add_vc</strong></span> ivc_a, ivc_b</pre>
<pre class="synopsis">imr <span class="command"><strong>la_i_add_mr</strong></span> imr_a, imr_b</pre>
<pre class="synopsis">imc <span class="command"><strong>la_k_add_mc</strong></span> imc_a, imc_b</pre>
<pre class="synopsis">ivr <span class="command"><strong>la_i_subtract_vr</strong></span> ivr_a, ivr_b</pre>
<pre class="synopsis">ivc <span class="command"><strong>la_k_subtract_vc</strong></span> ivc_a, ivc_b</pre>
<pre class="synopsis">imr <span class="command"><strong>la_i_subtract_mr</strong></span> imr_a, imr_b</pre>
<pre class="synopsis">imc <span class="command"><strong>la_k_subtract_mc</strong></span> imc_a, imc_b</pre>
<pre class="synopsis">ivr <span class="command"><strong>la_i_multiply_vr</strong></span> ivr_a, ivr_b</pre>
<pre class="synopsis">ivc <span class="command"><strong>la_k_multiply_vc</strong></span> ivc_a, ivc_b</pre>
<pre class="synopsis">imr <span class="command"><strong>la_i_multiply_mr</strong></span> imr_a, imr_b</pre>
<pre class="synopsis">imc <span class="command"><strong>la_k_multiply_mc</strong></span> imc_a, imc_b</pre>
<pre class="synopsis">ivr <span class="command"><strong>la_i_divide_vr</strong></span> ivr_a, ivr_b</pre>
<pre class="synopsis">ivc <span class="command"><strong>la_k_divide_vc</strong></span> ivc_a, ivc_b</pre>
<pre class="synopsis">imr <span class="command"><strong>la_i_divide_mr</strong></span> imr_a, imr_b</pre>
<pre class="synopsis">imc <span class="command"><strong>la_k_divide_mc</strong></span> imc_a, imc_b</pre>
<h4><a id="idp10500592"></a>
Inner Products
</h4>
<pre class="synopsis">ir <span class="command"><strong>la_i_dot_vr</strong></span> ivr_a, ivr_b</pre>
<pre class="synopsis">kr <span class="command"><strong>la_k_dot_vr</strong></span> ivr_a, ivr_b</pre>
<pre class="synopsis">ir, ii <span class="command"><strong>la_i_dot_vc</strong></span> ivc_a, ivc_b</pre>
<pre class="synopsis">kr, ki <span class="command"><strong>la_k_dot_vc</strong></span> ivc_a, ivc_b</pre>
<pre class="synopsis">imr <span class="command"><strong>la_i_dot_mr</strong></span> imr_a, imr_b</pre>
<pre class="synopsis">imr <span class="command"><strong>la_k_dot_mr</strong></span> imr_a, imr_b</pre>
<pre class="synopsis">imc <span class="command"><strong>la_i_dot_mc</strong></span> imc_a, imc_b</pre>
<pre class="synopsis">imc <span class="command"><strong>la_k_dot_mc</strong></span> imc_a, imc_b</pre>
<pre class="synopsis">ivr <span class="command"><strong>la_i_dot_mr_vr</strong></span> imr_a, ivr_b</pre>
<pre class="synopsis">ivr <span class="command"><strong>la_k_dot_mr_vr</strong></span> imr_a, ivr_b</pre>
<pre class="synopsis">ivc <span class="command"><strong>la_i_dot_mc_vc</strong></span> imc_a, ivc_b</pre>
<pre class="synopsis">ivc <span class="command"><strong>la_k_dot_mc_vc</strong></span> imc_a, ivc_b</pre>
<h4><a id="idp10509872"></a>
Matrix Inversion
</h4>
<pre class="synopsis">imr, icondition <span class="command"><strong>la_i_invert_mr</strong></span> imr</pre>
<pre class="synopsis">imr, kcondition <span class="command"><strong>la_k_invert_mr</strong></span> imr</pre>
<pre class="synopsis">imc, icondition <span class="command"><strong>la_i_invert_mc</strong></span> imc</pre>
<pre class="synopsis">imc, kcondition <span class="command"><strong>la_k_invert_mc</strong></span> imc</pre>
<h4><a id="idp10513080"></a>
Matrix Decompositions and Solvers
</h4>
<pre class="synopsis">ivr <span class="command"><strong>la_i_upper_solve_mr</strong></span> imr [, j_1_diagonal]</pre>
<pre class="synopsis">ivr <span class="command"><strong>la_k_upper_solve_mr</strong></span> imr [, j_1_diagonal]</pre>
<pre class="synopsis">ivc <span class="command"><strong>la_i_upper_solve_mc</strong></span> imc [, j_1_diagonal]</pre>
<pre class="synopsis">ivc <span class="command"><strong>la_k_upper_solve_mc</strong></span> imc [, j_1_diagonal]</pre>
<pre class="synopsis">ivr <span class="command"><strong>la_i_lower_solve_mr</strong></span> imr [, j_1_diagonal]</pre>
<pre class="synopsis">ivr <span class="command"><strong>la_k_lower_solve_mr</strong></span> imr [, j_1_diagonal]</pre>
<pre class="synopsis">ivc <span class="command"><strong>la_i_lower_solve_mc</strong></span> imc [, j_1_diagonal]</pre>
<pre class="synopsis">ivc <span class="command"><strong>la_k_lower_solve_mc</strong></span> imc [, j_1_diagonal]</pre>
<pre class="synopsis">imr, ivr_pivot, isize <span class="command"><strong>la_i_lu_factor_mr</strong></span> imr</pre>
<pre class="synopsis">imr, ivr_pivot, ksize <span class="command"><strong>la_k_lu_factor_mr</strong></span> imr</pre>
<pre class="synopsis">imc, ivr_pivot, isize <span class="command"><strong>la_i_lu_factor_mc</strong></span> imc</pre>
<pre class="synopsis">imc, ivr_pivot, ksize <span class="command"><strong>la_k_lu_factor_mc</strong></span> imc</pre>
<pre class="synopsis">ivr_x <span class="command"><strong>la_i_lu_solve_mr</strong></span> imr, ivr_b</pre>
<pre class="synopsis">ivr_x <span class="command"><strong>la_k_lu_solve_mr</strong></span> imr, ivr_b</pre>
<pre class="synopsis">ivc_x <span class="command"><strong>la_i_lu_solve_mc</strong></span> imc, ivc_b</pre>
<pre class="synopsis">ivc_x <span class="command"><strong>la_k_lu_solve_mc</strong></span> imc, ivc_b</pre>
<pre class="synopsis">imr_q, imr_r <span class="command"><strong>la_i_qr_factor_mr</strong></span> imr</pre>
<pre class="synopsis">imr_q, imr_r <span class="command"><strong>la_k_qr_factor_mr</strong></span> imr</pre>
<pre class="synopsis">imc_q, imc_r <span class="command"><strong>la_i_qr_factor_mc</strong></span> imc</pre>
<pre class="synopsis">imc_q, imc_r <span class="command"><strong>la_k_qr_factor_mc</strong></span> imc</pre>
<pre class="synopsis">ivr_eig_vals <span class="command"><strong>la_i_qr_eigen_mr</strong></span> imr, i_tolerance</pre>
<pre class="synopsis">ivr_eig_vals <span class="command"><strong>la_k_qr_eigen_mr</strong></span> imr, k_tolerance</pre>
<pre class="synopsis">ivr_eig_vals <span class="command"><strong>la_i_qr_eigen_mc</strong></span> imc, i_tolerance</pre>
<pre class="synopsis">ivr_eig_vals <span class="command"><strong>la_k_qr_eigen_mc</strong></span> imc, k_tolerance</pre>
<div class="warning" title="Warning" style="margin-left: 0.5in; margin-right: 0.5in;">
<table border="0" summary="Warning">
<tr>
<td rowspan="2" align="center" valign="top" width="25">
<img alt="[Warning]" src="images/warning.png" />
</td>
<th align="left">Warning</th>
</tr>
<tr>
<td align="left" valign="top">
Matrix must be Hermitian in order to compute eigenvectors.
</td>
</tr>
</table>
</div>
<pre class="synopsis">ivr_eig_vals, imr_eig_vecs <span class="command"><strong>la_i_qr_sym_eigen_mr</strong></span> imr, i_tolerance</pre>
<pre class="synopsis">ivr_eig_vals, imr_eig_vecs <span class="command"><strong>la_k_qr_sym_eigen_mr</strong></span> imr, k_tolerance</pre>
<pre class="synopsis">ivc_eig_vals, imc_eig_vecs <span class="command"><strong>la_i_qr_sym_eigen_mc</strong></span> imc, i_tolerance</pre>
<pre class="synopsis">ivc_eig_vals, imc_eig_vecs <span class="command"><strong>la_k_qr_sym_eigen_mc</strong></span> imc, k_tolerance</pre>
</div>
<div class="refsect1" title="Credits">
<a id="idp10534776"></a>
<h2>Credits</h2>
<p>Michael Gogins</p>
<p>New in Csound version 5.09</p>
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