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Eigh eigenvectors are inverted

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4/5
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3-5 天
新手友好度
35/100
Issue 类型
缺陷
描述清晰度
基本清楚
活跃度
停滞
技术栈
rust
领域
data

调研方向

Start by running the MWE from main.rs with the dependencies in Cargo.toml and compare the results for UPLO::Lower and UPLO::Upper. Trace the eigh entry point and its LAPACK argument handling; done means A @ eigenvectors matches eigenvectors @ eigenvalues without reversing rows and the reported UPLO behavior is consistent.

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描述

Related: #307

MWE:

Cargo.toml:

[package]
name = "EighBug"
version = "0.1.0"
edition = "2021"

# See more keys and their definitions at https://doc.rust-lang.org/cargo/reference/manifest.html

[dependencies]
# ndarray
ndarray = "0.15.6"
# ndarray_linalg
ndarray-linalg = { version = "0.16.0", features = ["openblas-static"] }
# num
num = "0.4.1"
# rustfmt
rustfmt = "0.10.0"

main.rs:

use ndarray::*;
use ndarray_linalg::*;
use num::complex::Complex64;

fn main() {
    let A = arr2(&[
        [
            Complex64::new(3., 0.),
            Complex64::new(2., 2.),
            Complex64::new(2., 2.),
        ],
        [
            Complex64::new(2., -2.),
            Complex64::new(3., 0.),
            Complex64::new(2., 2.),
        ],
        [
            Complex64::new(2., -2.),
            Complex64::new(2., -2.),
            Complex64::new(3., 0.),
        ],
    ]);

    println!("A = \n{}", A);

    let (eigenvalues, eigenvectors) = A.eigh(UPLO::Lower).unwrap();

    println!(
        "eigenvalues = \n{}",
        eigenvalues.mapv(|x| format!("{:.3}", x))
    );

    println!(
        "eigenvectors = \n{}",
        eigenvectors.mapv(|x| format!("{:.3}", x))
    );

    println!(
        "A @ eigenvectors = \n{}",
        A.dot(&eigenvectors).mapv(|x| format!("{:.3}", x))
    );

    println!(
        "eigenvectors @ eigenvalues = \n{}",
        eigenvectors
            .dot(&Array2::from_diag(
                &eigenvalues.mapv(|x| Complex64::from(x))
            ))
            .mapv(|x| format!("{:.3}", x))
    );

    println!("==============================");
    println!("WITH REVERSED ROWS");

    let eigenvectors = eigenvectors.slice(s![..;-1, ..]).to_owned();
    println!(
        "eigenvectors = \n{}",
        eigenvectors.mapv(|x| format!("{:.3}", x))
    );

    println!(
        "A @ eigenvectors = \n{}",
        A.dot(&eigenvectors).mapv(|x| format!("{:.3}", x))
    );

    println!(
        "eigenvectors @ eigenvalues = \n{}",
        eigenvectors
            .dot(&Array2::from_diag(
                &eigenvalues.mapv(|x| Complex64::from(x))
            ))
            .mapv(|x| format!("{:.3}", x))
    );
}

Output

A = 
[[3+0i, 2+2i, 2+2i],
 [2-2i, 3+0i, 2+2i],
 [2-2i, 2-2i, 3+0i]]
eigenvalues = 
[-1.000, 1.536, 8.464]
eigenvectors = 
[[-0.577+0.000i, -0.577+0.000i, -0.577+0.000i],
 [-0.000+0.577i, 0.500-0.289i, -0.500-0.289i],
 [0.577-0.000i, -0.289+0.500i, -0.289-0.500i]]
A @ eigenvectors = 
[[-1.732+2.309i, -1.732+0.845i, -1.732-3.155i],
 [0.000+4.041i, -1.232+0.711i, -2.232-1.289i],
 [1.732+2.309i, -1.598+1.077i, -3.598+0.077i]]
eigenvectors @ eigenvalues = 
[[0.577+0.000i, -0.887+0.000i, -4.887+0.000i],
 [0.000-0.577i, 0.768-0.443i, -4.232-2.443i],
 [-0.577+0.000i, -0.443+0.768i, -2.443-4.232i]]
==============================
WITH REVERSED ROWS
eigenvectors = 
[[0.577-0.000i, -0.289+0.500i, -0.289-0.500i],
 [-0.000+0.577i, 0.500-0.289i, -0.500-0.289i],
 [-0.577+0.000i, -0.577+0.000i, -0.577+0.000i]]
A @ eigenvectors = 
[[-0.577+0.000i, -0.443+0.768i, -2.443-4.232i],
 [0.000-0.577i, 0.768-0.443i, -4.232-2.443i],
 [0.577+0.000i, -0.887-0.000i, -4.887+0.000i]]
eigenvectors @ eigenvalues = 
[[-0.577+0.000i, -0.443+0.768i, -2.443-4.232i],
 [0.000-0.577i, 0.768-0.443i, -4.232-2.443i],
 [0.577+0.000i, -0.887+0.000i, -4.887+0.000i]]

Expected output

A @ eigenvectors should equal eigenvectors @ eigenvalues without needing to invert the row ordering.

This could very likely be a misunderstanding on my end. But it may be related to the reversal of UPLO::Upper and UPLO::Lower. For example, for the array:

A = [[3+0i, 2+2i, 2+2i],
     [   0, 3+0i, 2+2i],
     [   0,    0, 3+0i]]

eigh(UPLO::upper) returns [3, 3, 3] for the eigenvalues, which is the expected result for eigh(UPLO::lower).

主要语言
Rust
星标
452
派生
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