feat: add stats/strided/scovarmtk

lib/node_modules/@stdlib/stats/strided/scovarmtk/README.md

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+<!--
+
+@license Apache-2.0
+
+Copyright (c) 2025 The Stdlib Authors.
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+   http://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+
+-->
+
+<!-- lint disable maximum-heading-length -->
+
+# scovarmtk
+
+> Calculate the [covariance][covariance] of two single-precision floating-point strided arrays provided known means and using a one-pass textbook algorithm.
+
+<section class="intro">
+
+The population [covariance][covariance] of two finite size populations of size `N` is given by
+
+<!-- <equation class="equation" label="eq:population_covariance" align="center" raw="\operatorname{\mathrm{cov_N}} = \frac{1}{N} \sum_{i=0}^{N-1} (x_i - \mu_x)(y_i - \mu_y)" alt="Equation for the population covariance."> -->
+
+```math
+\mathop{\mathrm{cov_N}} = \frac{1}{N} \sum_{i=0}^{N-1} (x_i - \mu_x)(y_i - \mu_y)
+```
+
+<!-- </equation> -->
+
+where the population means are given by
+
+<!-- <equation class="equation" label="eq:population_mean_for_x" align="center" raw="\mu_x = \frac{1}{N} \sum_{i=0}^{N-1} x_i" alt="Equation for the population mean for first array."> -->
+
+```math
+\mu_x = \frac{1}{N} \sum_{i=0}^{N-1} x_i
+```
+
+<!-- </equation> -->
+
+and
+
+<!-- <equation class="equation" label="eq:population_mean_for_y" align="center" raw="\mu_y = \frac{1}{N} \sum_{i=0}^{N-1} y_i" alt="Equation for the population mean for second array."> -->
+
+```math
+\mu_y = \frac{1}{N} \sum_{i=0}^{N-1} y_i
+```
+
+<!-- </equation> -->
+
+Often in the analysis of data, the true population [covariance][covariance] is not known _a priori_ and must be estimated from samples drawn from population distributions. If one attempts to use the formula for the population [covariance][covariance], the result is biased and yields a **biased sample covariance**. To compute an **unbiased sample covariance** for samples of size `n`,
+
+<!-- <equation class="equation" label="eq:unbiased_sample_covariance" align="center" raw="\operatorname{\mathrm{cov_n}} = \frac{1}{n-1} \sum_{i=0}^{n-1} (x_i - \bar{x}_n)(y_i - \bar{y}_n)" alt="Equation for computing an unbiased sample variance."> -->
+
+```math
+\mathop{\mathrm{cov_n}} = \frac{1}{n-1} \sum_{i=0}^{n-1} (x_i - \bar{x}_n)(y_i - \bar{y}_n)
+```
+
+<!-- </equation> -->
+
+where sample means are given by
+
+<!-- <equation class="equation" label="eq:sample_mean_for_x" align="center" raw="\bar{x} = \frac{1}{n} \sum_{i=0}^{n-1} x_i" alt="Equation for the sample mean for first array."> -->
+
+```math
+\bar{x} = \frac{1}{n} \sum_{i=0}^{n-1} x_i
+```
+
+<!-- </equation> -->
+
+and
+
+<!-- <equation class="equation" label="eq:sample_mean_for_y" align="center" raw="\bar{y} = \frac{1}{n} \sum_{i=0}^{n-1} y_i" alt="Equation for the sample mean for second array."> -->
+
+```math
+\bar{y} = \frac{1}{n} \sum_{i=0}^{n-1} y_i
+```
+
+<!-- </equation> -->
+
+The use of the term `n-1` is commonly referred to as Bessel's correction. Depending on the characteristics of the population distributions, other correction factors (e.g., `n-1.5`, `n+1`, etc) can yield better estimators.
+
+</section>
+
+<!-- /.intro -->
+
+<section class="usage">
+
+## Usage
+
+```javascript
+var scovarmtk = require( '@stdlib/stats/strided/scovarmtk' );
+```
+
+#### scovarmtk( N, correction, meanx, x, strideX, meany, y, strideY )
+
+Computes the [covariance][covariance] of two single-precision floating-point strided arrays provided known means and using a one-pass textbook algorithm.
+
+```javascript
+var Float32Array = require( '@stdlib/array/float32' );
+
+var x = new Float32Array( [ 1.0, -2.0, 2.0 ] );
+var y = new Float32Array( [ 2.0, -2.0, 1.0 ] );
+
+var v = scovarmtk( x.length, 1, 1.0/3.0, x, 1, 1.0/3.0, y, 1 );
+// returns ~3.8333
+```
+
+The function has the following parameters:
+
+-   **N**: number of indexed elements.
+-   **correction**: degrees of freedom adjustment. Setting this parameter to a value other than `0` has the effect of adjusting the divisor during the calculation of the [covariance][covariance] according to `N-c` where `c` corresponds to the provided degrees of freedom adjustment. When computing the population [covariance][covariance], setting this parameter to `0` is the standard choice (i.e., the provided arrays contain data constituting entire populations). When computing the unbiased sample [covariance][covariance], setting this parameter to `1` is the standard choice (i.e., the provided arrays contain data sampled from larger populations; this is commonly referred to as Bessel's correction).
+-   **meanx**: mean of `x`.
+-   **x**: first input [`Float32Array`][@stdlib/array/float32].
+-   **strideX**: stride length for `x`.
+-   **meany**: mean of `y`.
+-   **y**: second input [`Float32Array`][@stdlib/array/float32].
+-   **strideY**: stride length for `y`.
+
+The `N` and stride parameters determine which elements in the strided arrays are accessed at runtime. For example, to compute the [covariance][covariance] of every other element in `x` and `y`,
+
+```javascript
+var Float32Array = require( '@stdlib/array/float32' );
+
+var x = new Float32Array( [ 1.0, 2.0, 2.0, -7.0, -2.0, 3.0, 4.0, 2.0 ] );
+var y = new Float32Array( [ 2.0, 1.0, 2.0, 1.0, -2.0, 2.0, 3.0, 4.0 ] );
+
+var v = scovarmtk( 4, 1, 1.25, x, 2, 1.25, y, 2 );
+// returns 6.0
+```
+
+Note that indexing is relative to the first index. To introduce an offset, use [`typed array`][mdn-typed-array] views.
+
+<!-- eslint-disable stdlib/capitalized-comments -->
+
+```javascript
+var Float32Array = require( '@stdlib/array/float32' );
+
+var x0 = new Float32Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );
+var y0 = new Float32Array( [ 2.0, -2.0, 2.0, 1.0, -2.0, 4.0, 3.0, 2.0 ] );
+
+var x1 = new Float32Array( x0.buffer, x0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
+var y1 = new Float32Array( y0.buffer, y0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
+
+var v = scovarmtk( 4, 1, 1.25, x1, 2, 1.25, y1, 2 );
+// returns ~1.9167
+```
+
+#### scovarmtk.ndarray( N, correction, meanx, x, strideX, offsetX, meany, y, strideY, offsetY )
+
+Computes the [covariance][covariance] of two single-precision floating-point strided arrays provided known means and using a one-pass textbook algorithm and alternative indexing semantics.
+
+```javascript
+var Float32Array = require( '@stdlib/array/float32' );
+
+var x = new Float32Array( [ 1.0, -2.0, 2.0 ] );
+var y = new Float32Array( [ 2.0, -2.0, 1.0 ] );
+
+var v = scovarmtk.ndarray( x.length, 1, 1.0/3.0, x, 1, 0, 1.0/3.0, y, 1, 0 );
+// returns ~3.8333
+```
+
+The function has the following additional parameters:
+
+-   **offsetX**: starting index for `x`.
+-   **offsetY**: starting index for `y`.
+
+While [`typed array`][mdn-typed-array] views mandate a view offset based on the underlying buffer, the offset parameters support indexing semantics based on starting indices. For example, to calculate the [covariance][covariance] for every other element in `x` and `y` starting from the second element
+
+```javascript
+var Float32Array = require( '@stdlib/array/float32' );
+
+var x = new Float32Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );
+var y = new Float32Array( [ -7.0, 2.0, 2.0, 1.0, -2.0, 2.0, 3.0, 4.0 ] );
+
+var v = scovarmtk.ndarray( 4, 1, 1.25, x, 2, 1, 1.25, y, 2, 1 );
+// returns 6.0
+```
+
+</section>
+
+<!-- /.usage -->
+
+<section class="notes">
+
+## Notes
+
+-   If `N <= 0`, both functions return `NaN`.
+-   If `N - c` is less than or equal to `0` (where `c` corresponds to the provided degrees of freedom adjustment), both functions return `NaN`.
+
+</section>
+
+<!-- /.notes -->
+
+<section class="examples">
+
+## Examples
+
+<!-- eslint no-undef: "error" -->
+
+```javascript
+var discreteUniform = require( '@stdlib/random/array/discrete-uniform' );
+var scovarmtk = require( '@stdlib/stats/strided/scovarmtk' );
+
+var opts = {
+    'dtype': 'float32'
+};
+var x = discreteUniform( 10, -50, 50, opts );
+console.log( x );
+
+var y = discreteUniform( 10, -50, 50, opts );
+console.log( y );
+
+var v = scovarmtk( x.length, 1, 0.0, x, 1, 0.0, y, 1 );
+console.log( v );
+```
+
+</section>
+
+<!-- /.examples -->
+
+<!-- C interface documentation. -->
+
+* * *
+
+<section class="c">
+
+## C APIs
+
+<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->
+
+<section class="intro">
+
+</section>
+
+<!-- /.intro -->
+
+<!-- C usage documentation. -->
+
+<section class="usage">
+
+### Usage
+
+```c
+#include "stdlib/stats/strided/scovarmtk.h"
+```
+
+#### stdlib_strided_scovarmtk( N, correction, meanx, \*X, strideX, meany, \*Y, strideY )
+
+Computes the [covariance][covariance] of two single-precision floating-point strided arrays provided known means and using a one-pass textbook algorithm.
+
+```c
+const float x[] = { 1.0f, -2.0f, 2.0f };
+const float y[] = { 2.0f, -2.0f, 1.0f };
+
+float v = stdlib_strided_scovarmtk( 3, 1.0f, 1.0f/3.0f, x, 1, 1.0f/3.0f, y, 1 );
+// returns ~7.6667f
+```
+
+The function accepts the following arguments:
+
+-   **N**: `[in] CBLAS_INT` number of indexed elements.
+-   **correction**: `[in] float` degrees of freedom adjustment. Setting this parameter to a value other than `0` has the effect of adjusting the divisor during the calculation of the [covariance][covariance] according to `N-c` where `c` corresponds to the provided degrees of freedom adjustment. When computing the population [covariance][covariance], setting this parameter to `0` is the standard choice (i.e., the provided arrays contain data constituting entire populations). When computing the unbiased sample [covariance][covariance], setting this parameter to `1` is the standard choice (i.e., the provided arrays contain data sampled from larger populations; this is commonly referred to as Bessel's correction).
+-   **meanx**: `[in] float` mean of `X`.
+-   **X**: `[in] float*` first input array.
+-   **strideX**: `[in] CBLAS_INT` stride length for `X`.
+-   **meany**: `[in] float` mean of `Y`.
+-   **Y**: `[in] float*` second input array.
+-   **strideY**: `[in] CBLAS_INT` stride length for `Y`.
+
+```c
+float stdlib_strided_scovarmtk( const CBLAS_INT N, const float correction, const float meanx, const float *X, const CBLAS_INT strideX, const float meany, const float *Y, const CBLAS_INT strideY );
+```
+
+#### stdlib_strided_scovarmtk_ndarray( N, correction, meanx, \*X, strideX, offsetX, meany, \*Y, strideY, offsetY )
+
+Computes the [covariance][covariance] of two single-precision floating-point strided arrays provided known means and using a one-pass textbook algorithm and alternative indexing semantics.
+
+```c
+const float x[] = { 1.0f, -2.0f, 2.0f };
+const float y[] = { 2.0f, -2.0f, 1.0f };
+
+float v = stdlib_strided_scovarmtk_ndarray( 3, 1.0f, 1.0f/3.0f, x, 1, 0, 1.0f/3.0f, y, 1, 0 );
+// returns ~7.6667f
+```
+
+The function accepts the following arguments:
+
+-   **N**: `[in] CBLAS_INT` number of indexed elements.
+-   **correction**: `[in] float` degrees of freedom adjustment. Setting this parameter to a value other than `0` has the effect of adjusting the divisor during the calculation of the [covariance][covariance] according to `N-c` where `c` corresponds to the provided degrees of freedom adjustment. When computing the population [covariance][covariance], setting this parameter to `0` is the standard choice (i.e., the provided arrays contain data constituting entire populations). When computing the unbiased sample [covariance][covariance], setting this parameter to `1` is the standard choice (i.e., the provided arrays contain data sampled from larger populations; this is commonly referred to as Bessel's correction).
+-   **meanx**: `[in] float` mean of `X`.
+-   **X**: `[in] float*` first input array.
+-   **strideX**: `[in] CBLAS_INT` stride length for `X`.
+-   **offsetX**: `[in] CBLAS_INT` starting index for `X`.
+-   **meany**: `[in] float` mean of `Y`.
+-   **Y**: `[in] float*` second input array.
+-   **strideY**: `[in] CBLAS_INT` stride length for `Y`.
+-   **offsetY**: `[in] CBLAS_INT` starting index for `Y`.
+
+```c
+float stdlib_strided_scovarmtk_ndarray( const CBLAS_INT N, const float correction, const float meanx, const float *X, const CBLAS_INT strideX, const CBLAS_INT offsetX, const float meany, const float *Y, const CBLAS_INT strideY, const CBLAS_INT offsetY );
+```
+
+</section>
+
+<!-- /.usage -->
+
+<!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
+
+<section class="notes">
+
+</section>
+
+<!-- /.notes -->
+
+<!-- C API usage examples. -->
+
+<section class="examples">
+
+### Examples
+
+```c
+#include "stdlib/stats/strided/scovarmtk.h"
+#include <stdio.h>
+
+int main( void ) {
+    // Create a strided array:
+    const float x[] = { 1.0f, 2.0f, 3.0f, 4.0f, 5.0f, 6.0f, 7.0f, 8.0f };
+
+    // Specify the number of elements:
+    const int N = 4;
+
+    // Specify the stride length:
+    const int strideX = 2;
+
+    // Compute the covariance of `x` with itself:
+    float v = stdlib_strided_scovarmtk( N, 1.0f, 4.5f, x, strideX, 4.5f, x, -strideX );
+
+    // Print the result:
+    printf( "covariance: %f\n", v );
+}
+```
+
+</section>
+
+<!-- /.examples -->
+
+</section>
+
+<!-- /.c -->
+
+<section class="references">
+
+</section>
+
+<!-- /.references -->
+
+<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->
+
+<section class="related">
+
+</section>
+
+<!-- /.related -->
+
+<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->
+
+<section class="links">
+
+[covariance]: https://en.wikipedia.org/wiki/Covariance
+
+[@stdlib/array/float32]: https://github.com/stdlib-js/stdlib/tree/develop/lib/node_modules/%40stdlib/array/float32
+
+[mdn-typed-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/TypedArray
+
+</section>
+
+<!-- /.links -->