bdpseudoinv

bdpseudoinv

LINEAR_ALGEBRA

1 Description

Computes the Moore-Penrose pseudoinverse of a matrix using SVD decomposition. This implementation handles both square and rectangular matrices, and provides numerically stable results even for singular or near-singular matrices.

2 Usage

bdpseudoinv(X, threads = NULL)

3 Arguments

Parameter	Description
`X`	Numeric matrix or vector to be pseudoinverted.
`threads`	Optional integer. Number of threads for parallel computation. If NULL, uses maximum available threads.

4 Value

The pseudoinverse matrix of X.

5 Details

The Moore-Penrose pseudoinverse (denoted A+) of a matrix A is computed using Singular Value Decomposition (SVD).

For a matrix A = USigmaV^T (where ^T denotes transpose), the pseudoinverse is computed as:

where Sigma+ is obtained by taking the reciprocal of non-zero singular values.

6 Examples

library(BigDataStatMeth)

# Create a singular matrix
X <- matrix(c(1,2,3,2,4,6), 2, 3)  # rank-deficient matrix

# Compute pseudoinverse
X_pinv <- bdpseudoinv(X)

# Verify Moore-Penrose conditions
# 1. X %*% X_pinv %*% X = X
all.equal(X %*% X_pinv %*% X, X)

# 2. X_pinv %*% X %*% X_pinv = X_pinv
all.equal(X_pinv %*% X %*% X_pinv, X_pinv)

7 See Also

bdpseudoinv_hdf5 for HDF5-stored matrices
bdSVD_hdf5 for singular value decomposition

---
title: "bdpseudoinv"
subtitle: "bdpseudoinv"
---

<span class="category-badge linear_algebra">LINEAR_ALGEBRA</span>

## Description

Computes the Moore-Penrose pseudoinverse of a matrix using SVD decomposition.
This implementation handles both square and rectangular matrices, and provides
numerically stable results even for singular or near-singular matrices.

## Usage

```r
bdpseudoinv(X, threads = NULL)
```
## Arguments

::: {.param-table}
| Parameter | Description |
|-----------|-------------|
| `X` | Numeric matrix or vector to be pseudoinverted. |
| `threads` | Optional integer. Number of threads for parallel computation.   If NULL, uses maximum available threads.  |
:::

## Value

::: {.return-value}
The pseudoinverse matrix of X.

:::
## Details

The Moore-Penrose pseudoinverse (denoted A+) of a matrix A is computed using 
Singular Value Decomposition (SVD). 

For a matrix A = U*Sigma*V^T (where ^T denotes transpose), the pseudoinverse is 
computed as:

\deqn{A^+ = V \Sigma^+ U^T}

where Sigma+ is obtained by taking the reciprocal of non-zero singular values.

## Examples
```{r}
#| eval: false
#| warning: false

library(BigDataStatMeth)

# Create a singular matrix
X <- matrix(c(1,2,3,2,4,6), 2, 3)  # rank-deficient matrix

# Compute pseudoinverse
X_pinv <- bdpseudoinv(X)

# Verify Moore-Penrose conditions
# 1. X %*% X_pinv %*% X = X
all.equal(X %*% X_pinv %*% X, X)

# 2. X_pinv %*% X %*% X_pinv = X_pinv
all.equal(X_pinv %*% X %*% X_pinv, X_pinv)

```
## See Also

::: {.see-also}
- [bdpseudoinv_hdf5](../hdf5_algebra/bdpseudoinv_hdf5.html) for HDF5-stored matrices
- [bdSVD_hdf5](../hdf5_algebra/bdSVD_hdf5.html) for singular value decomposition

:::