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use crate::matrix::*;
use russell_openblas::*;
/// Performs the eigen-decomposition of a square matrix
///
/// Computes the eigenvalues `l` and right eigenvectors `v`, such that:
///
/// ```text
/// a ⋅ vj = lj ⋅ vj
/// ```
///
/// where `lj` is the component j of `l` and `vj` is the column j of `v`.
///
/// # Output
///
/// * `l_real` -- (m) eigenvalues; real part
/// * `l_imag` -- (m) eigenvalues; imaginary part
/// * `v_real` -- (m,m) **right** eigenvectors (as columns); real part
/// * `v_imag` -- (m,m) **right** eigenvectors (as columns); imaginary part
///
/// # Input
///
/// * `a` -- (m,m) general matrix [will be modified]
///
/// # Note
///
/// * The matrix `a` will be modified
///
/// # Similarity transformation
///
/// The eigen-decomposition leads to a similarity transformation like so:
///
/// ```text
/// a = v⋅λ⋅v⁻¹
/// ```
///
/// where `v` is a matrix whose columns are the m linearly independent eigenvectors of `a`,
/// and `λ` is a matrix whose diagonal are the eigenvalues of `a`. Thus, the following is valid:
///
/// ```text
/// a⋅v = v⋅λ
/// ```
///
/// Let us define the error `err` as follows:
///
/// ```text
/// err := a⋅v - v⋅λ
/// ```
///
/// # Example
///
/// ```
/// # fn main() -> Result<(), &'static str> {
/// // import
/// use russell_lab::*;
/// use russell_chk::*;
///
/// // set matrix
/// let data: &[&[f64]] = &[
/// &[2.0, 0.0, 0.0],
/// &[0.0, 3.0, 4.0],
/// &[0.0, 4.0, 9.0],
/// ];
/// let mut a = Matrix::from(data)?;
///
/// // allocate output arrays
/// let m = a.nrow();
/// let mut l_real = vec![0.0; m];
/// let mut l_imag = vec![0.0; m];
/// let mut v_real = Matrix::new(m, m);
/// let mut v_imag = Matrix::new(m, m);
///
/// // perform the eigen-decomposition
/// eigen_decomp(
/// &mut l_real,
/// &mut l_imag,
/// &mut v_real,
/// &mut v_imag,
/// &mut a,
/// )?;
///
/// // check results
/// let l_real_correct = "[11.0, 1.0, 2.0]";
/// let l_imag_correct = "[0.0, 0.0, 0.0]";
/// let v_real_correct = "┌ ┐\n\
/// │ 0.000 0.000 1.000 │\n\
/// │ 0.447 0.894 0.000 │\n\
/// │ 0.894 -0.447 0.000 │\n\
/// └ ┘";
/// let v_imag_correct = "┌ ┐\n\
/// │ 0 0 0 │\n\
/// │ 0 0 0 │\n\
/// │ 0 0 0 │\n\
/// └ ┘";
/// assert_eq!(format!("{:?}", l_real), l_real_correct);
/// assert_eq!(format!("{:?}", l_imag), l_imag_correct);
/// assert_eq!(format!("{:.3}", v_real), v_real_correct);
/// assert_eq!(format!("{}", v_imag), v_imag_correct);
///
/// // check eigen-decomposition (similarity transformation) of a
/// // symmetric matrix with real-only eigenvalues and eigenvectors
/// let a_copy = Matrix::from(data)?;
/// let lam = Matrix::diagonal(&l_real);
/// let mut a_v = Matrix::new(m, m);
/// let mut v_l = Matrix::new(m, m);
/// let mut err = Matrix::filled(m, m, f64::MAX);
/// mat_mat_mul(&mut a_v, 1.0, &a_copy, &v_real)?;
/// mat_mat_mul(&mut v_l, 1.0, &v_real, &lam)?;
/// add_matrices(&mut err, 1.0, &a_v, -1.0, &v_l)?;
/// assert_approx_eq!(err.norm(EnumMatrixNorm::Max), 0.0, 1e-15);
/// # Ok(())
/// # }
/// ```
pub fn eigen_decomp(
l_real: &mut [f64],
l_imag: &mut [f64],
v_real: &mut Matrix,
v_imag: &mut Matrix,
a: &mut Matrix,
) -> Result<(), &'static str> {
let (m, n) = (a.nrow, a.ncol);
if m != n {
return Err("matrix must be square");
}
let m_i32 = to_i32(m);
let mut v = vec![0.0; m * m];
let mut empty: Vec<f64> = Vec::new();
dgeev(
false,
true,
m_i32,
&mut a.data,
m_i32,
l_real,
l_imag,
&mut empty,
1,
&mut v,
m_i32,
)?;
dgeev_data(&mut v_real.data, &mut v_imag.data, l_imag, &v)?;
Ok(())
}
/// Performs the eigen-decomposition of a square matrix (left and right)
///
/// Computes the eigenvalues `l` and left eigenvectors `u`, such that:
///
/// ```text
/// ujᴴ ⋅ a = lj ⋅ ujᴴ
/// ```
///
/// where `lj` is the component j of `l` and `ujᴴ` is the column j of `uᴴ`,
/// with `uᴴ` being the conjugate-transpose of `u`.
///
/// Also, computes the right eigenvectors `v`, such that:
///
/// ```text
/// a ⋅ vj = lj ⋅ vj
/// ```
///
/// where `vj` is the column j of `v`.
///
/// # Output
///
/// * `l_real` -- (m) eigenvalues; real part
/// * `l_imag` -- (m) eigenvalues; imaginary part
/// * `u_real` -- (m,m) **left** eigenvectors (as columns); real part
/// * `u_imag` -- (m,m) **left** eigenvectors (as columns); imaginary part
/// * `v_real` -- (m,m) **right** eigenvectors (as columns); real part
/// * `v_imag` -- (m,m) **right** eigenvectors (as columns); imaginary part
///
/// # Input
///
/// * `a` -- (m,m) general matrix [will be modified]
///
/// # Note
///
/// * The matrix `a` will be modified
///
/// # Example
///
/// ```
/// # fn main() -> Result<(), &'static str> {
/// // import
/// use russell_lab::*;
///
/// // set matrix
/// let data: &[&[f64]] = &[
/// &[0.0, 1.0, 0.0],
/// &[0.0, 0.0, 1.0],
/// &[1.0, 0.0, 0.0],
/// ];
/// let mut a = Matrix::from(data)?;
///
/// // allocate output arrays
/// let m = a.nrow();
/// let mut l_real = vec![0.0; m];
/// let mut l_imag = vec![0.0; m];
/// let mut u_real = Matrix::new(m, m);
/// let mut u_imag = Matrix::new(m, m);
/// let mut v_real = Matrix::new(m, m);
/// let mut v_imag = Matrix::new(m, m);
///
/// // perform the eigen-decomposition
/// eigen_decomp_lr(
/// &mut l_real,
/// &mut l_imag,
/// &mut u_real,
/// &mut u_imag,
/// &mut v_real,
/// &mut v_imag,
/// &mut a,
/// )?;
///
/// // check results
/// let l_real_correct = "[-0.5, -0.5, 0.9999999999999998]";
/// let l_imag_correct = "[0.8660254037844389, -0.8660254037844389, 0.0]";
/// let u_real_correct = "┌ ┐\n\
/// │ -0.289 -0.289 -0.577 │\n\
/// │ 0.577 0.577 -0.577 │\n\
/// │ -0.289 -0.289 -0.577 │\n\
/// └ ┘";
/// let u_imag_correct = "┌ ┐\n\
/// │ -0.500 0.500 0.000 │\n\
/// │ 0.000 -0.000 0.000 │\n\
/// │ 0.500 -0.500 0.000 │\n\
/// └ ┘";
/// let v_real_correct = "┌ ┐\n\
/// │ 0.577 0.577 -0.577 │\n\
/// │ -0.289 -0.289 -0.577 │\n\
/// │ -0.289 -0.289 -0.577 │\n\
/// └ ┘";
/// let v_imag_correct = "┌ ┐\n\
/// │ 0.000 -0.000 0.000 │\n\
/// │ 0.500 -0.500 0.000 │\n\
/// │ -0.500 0.500 0.000 │\n\
/// └ ┘";
/// assert_eq!(format!("{:?}", l_real), l_real_correct);
/// assert_eq!(format!("{:?}", l_imag), l_imag_correct);
/// assert_eq!(format!("{:.3}", u_real), u_real_correct);
/// assert_eq!(format!("{:.3}", u_imag), u_imag_correct);
/// assert_eq!(format!("{:.3}", v_real), v_real_correct);
/// assert_eq!(format!("{:.3}", v_imag), v_imag_correct);
/// # Ok(())
/// # }
/// ```
pub fn eigen_decomp_lr(
l_real: &mut [f64],
l_imag: &mut [f64],
u_real: &mut Matrix,
u_imag: &mut Matrix,
v_real: &mut Matrix,
v_imag: &mut Matrix,
a: &mut Matrix,
) -> Result<(), &'static str> {
let (m, n) = (a.nrow, a.ncol);
if m != n {
return Err("matrix must be square");
}
let m_i32 = to_i32(m);
let mut u = vec![0.0; m * m];
let mut v = vec![0.0; m * m];
dgeev(
true,
true,
m_i32,
&mut a.data,
m_i32,
l_real,
l_imag,
&mut u,
m_i32,
&mut v,
m_i32,
)?;
dgeev_data_lr(
&mut u_real.data,
&mut u_imag.data,
&mut v_real.data,
&mut v_imag.data,
l_imag,
&u,
&v,
)?;
Ok(())
}
////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
#[cfg(test)]
mod tests {
use super::*;
use crate::EnumMatrixNorm;
use russell_chk::*;
fn check_real_eigen(data: &[&[f64]], v: &Matrix, l: &[f64]) -> Result<(), &'static str> {
let a = Matrix::from(data)?;
let m = a.nrow;
let lam = Matrix::diagonal(&l);
let mut a_v = Matrix::new(m, m);
let mut v_l = Matrix::new(m, m);
let mut err = Matrix::filled(m, m, f64::MAX);
mat_mat_mul(&mut a_v, 1.0, &a, &v)?;
mat_mat_mul(&mut v_l, 1.0, &v, &lam)?;
add_matrices(&mut err, 1.0, &a_v, -1.0, &v_l)?;
assert_approx_eq!(err.norm(EnumMatrixNorm::Max), 0.0, 1e-15);
Ok(())
}
#[test]
fn eigen_decomp_fails_on_non_square() {
let mut a = Matrix::new(3, 4);
let m = a.nrow;
let mut l_real = vec![0.0; m];
let mut l_imag = vec![0.0; m];
let mut v_real = Matrix::new(m, m);
let mut v_imag = Matrix::new(m, m);
assert_eq!(
eigen_decomp(&mut l_real, &mut l_imag, &mut v_real, &mut v_imag, &mut a),
Err("matrix must be square")
);
}
#[test]
fn eigen_decomp_lr_fails_on_non_square() {
let mut a = Matrix::new(3, 4);
let m = a.nrow;
let mut l_real = vec![0.0; m];
let mut l_imag = vec![0.0; m];
let mut u_real = Matrix::new(m, m);
let mut u_imag = Matrix::new(m, m);
let mut v_real = Matrix::new(m, m);
let mut v_imag = Matrix::new(m, m);
assert_eq!(
eigen_decomp_lr(
&mut l_real,
&mut l_imag,
&mut u_real,
&mut u_imag,
&mut v_real,
&mut v_imag,
&mut a,
),
Err("matrix must be square"),
);
}
#[test]
fn eigen_decomp_works() -> Result<(), &'static str> {
#[rustfmt::skip]
let data: &[&[f64]] = &[
&[0.0, 1.0, 0.0],
&[0.0, 0.0, 1.0],
&[1.0, 0.0, 0.0],
];
let mut a = Matrix::from(data)?;
let m = a.nrow;
let mut l_real = vec![0.0; m];
let mut l_imag = vec![0.0; m];
let mut v_real = Matrix::new(m, m);
let mut v_imag = Matrix::new(m, m);
eigen_decomp(&mut l_real, &mut l_imag, &mut v_real, &mut v_imag, &mut a)?;
let s3 = f64::sqrt(3.0);
let l_real_correct = &[-0.5, -0.5, 1.0];
let l_imag_correct = &[s3 / 2.0, -s3 / 2.0, 0.0];
#[rustfmt::skip]
let v_real_correct = Matrix::from(&[
&[ 1.0/s3, 1.0/s3, -1.0/s3],
&[-0.5/s3, -0.5/s3, -1.0/s3],
&[-0.5/s3, -0.5/s3, -1.0/s3],
])?;
#[rustfmt::skip]
let v_imag_correct = Matrix::from(&[
&[ 0.0, 0.0, 0.0],
&[ 0.5, -0.5, 0.0],
&[-0.5, 0.5, 0.0],
])?;
assert_vec_approx_eq!(l_real, l_real_correct, 1e-15);
assert_vec_approx_eq!(l_imag, l_imag_correct, 1e-15);
assert_vec_approx_eq!(v_real.data, v_real_correct.data, 1e-15);
assert_vec_approx_eq!(v_imag.data, v_imag_correct.data, 1e-15);
Ok(())
}
#[test]
fn eigen_decomp_rep_works() -> Result<(), &'static str> {
// rep: repeated eigenvalues
#[rustfmt::skip]
let data: &[&[f64]] = &[
&[2.0, 0.0, 0.0, 0.0],
&[1.0, 2.0, 0.0, 0.0],
&[0.0, 1.0, 3.0, 0.0],
&[0.0, 0.0, 1.0, 3.0],
];
let mut a = Matrix::from(data)?;
let m = a.nrow;
let mut l_real = vec![0.0; m];
let mut l_imag = vec![0.0; m];
let mut v_real = Matrix::new(m, m);
let mut v_imag = Matrix::new(m, m);
eigen_decomp(&mut l_real, &mut l_imag, &mut v_real, &mut v_imag, &mut a)?;
let l_real_correct = &[3.0, 3.0, 2.0, 2.0];
let l_imag_correct = &[0.0, 0.0, 0.0, 0.0];
let os3 = 1.0 / f64::sqrt(3.0);
#[rustfmt::skip]
let v_real_correct = Matrix::from(&[
&[ 0.0, 0.0, 0.0, 0.0],
&[ 0.0, 0.0, os3, -os3],
&[ 0.0, 0.0, -os3, os3],
&[ 1.0, -1.0, os3, -os3],
])?;
#[rustfmt::skip]
let v_imag_correct = Matrix::from(&[
&[0.0, 0.0, 0.0, 0.0],
&[0.0, 0.0, 0.0, 0.0],
&[0.0, 0.0, 0.0, 0.0],
&[0.0, 0.0, 0.0, 0.0],
])?;
assert_vec_approx_eq!(l_real, l_real_correct, 1e-15);
assert_vec_approx_eq!(l_imag, l_imag_correct, 1e-15);
assert_vec_approx_eq!(v_real.data, v_real_correct.data, 1e-15);
assert_vec_approx_eq!(v_imag.data, v_imag_correct.data, 1e-15);
check_real_eigen(&data, &v_real, &l_real)?;
Ok(())
}
#[test]
fn eigen_decomp_lr_works() -> Result<(), &'static str> {
#[rustfmt::skip]
let data: &[&[f64]] = &[
&[0.0, 1.0, 0.0],
&[0.0, 0.0, 1.0],
&[1.0, 0.0, 0.0],
];
let mut a = Matrix::from(data)?;
let m = a.nrow;
let mut l_real = vec![0.0; m];
let mut l_imag = vec![0.0; m];
let mut u_real = Matrix::new(m, m);
let mut u_imag = Matrix::new(m, m);
let mut v_real = Matrix::new(m, m);
let mut v_imag = Matrix::new(m, m);
eigen_decomp_lr(
&mut l_real,
&mut l_imag,
&mut u_real,
&mut u_imag,
&mut v_real,
&mut v_imag,
&mut a,
)?;
let s3 = f64::sqrt(3.0);
let l_real_correct = &[-0.5, -0.5, 1.0];
let l_imag_correct = &[s3 / 2.0, -s3 / 2.0, 0.0];
#[rustfmt::skip]
let u_real_correct = Matrix::from(&[
&[-0.5/s3, -0.5/s3, -1.0/s3],
&[ 1.0/s3, 1.0/s3, -1.0/s3],
&[-0.5/s3, -0.5/s3, -1.0/s3],
])?;
#[rustfmt::skip]
let u_imag_correct = Matrix::from(&[
&[-0.5, 0.5, 0.0],
&[ 0.0, 0.0, 0.0],
&[ 0.5, -0.5, 0.0],
])?;
#[rustfmt::skip]
let v_real_correct = Matrix::from(&[
&[ 1.0/s3, 1.0/s3, -1.0/s3],
&[-0.5/s3, -0.5/s3, -1.0/s3],
&[-0.5/s3, -0.5/s3, -1.0/s3],
])?;
#[rustfmt::skip]
let v_imag_correct = Matrix::from(&[
&[ 0.0, 0.0, 0.0],
&[ 0.5, -0.5, 0.0],
&[-0.5, 0.5, 0.0],
])?;
assert_vec_approx_eq!(l_real, l_real_correct, 1e-15);
assert_vec_approx_eq!(l_imag, l_imag_correct, 1e-15);
assert_vec_approx_eq!(u_real.data, u_real_correct.data, 1e-15);
assert_vec_approx_eq!(u_imag.data, u_imag_correct.data, 1e-15);
assert_vec_approx_eq!(v_real.data, v_real_correct.data, 1e-15);
assert_vec_approx_eq!(v_imag.data, v_imag_correct.data, 1e-15);
Ok(())
}
}