# EigenTridiagonalPosDefQ

Compute eigenvalues and eigenvectors of a symmetric positive definite tridiagonal matrix using the QR algorithm (LAPACK function [PTEQR](https://www.intel.com/content/www/us/en/docs/onemkl/developer-reference-fortran/2025-2/pteqr.html)). This method can be used to compute the eigenvectors of both the input tridiagonal matrix and the original positive definite symmetric matrix. This is achieved using the orthogonal matrix Q obtained during the reduction to tridiagonal form.

Computing for type matrix<double>

```
bool  matrix::EigenTridiagonalPosDefQ(
   ENUM_EIGTRIDIAG_Z     compv,              // compute eigenvectors or not
   matrix&               Q,                  // orthogonal matrix used in the reduction to tridiagonal form
   vector&               eigen_values,       // vector of computed eigenvalues
   matrix&               eigen_vectors       // matrix of computed eigenvectors
   );

```

Computing for type matrix<float>

```
bool  matrixf::EigenTridiagonalPosDefQ(
   ENUM_EIGTRIDIAG_Z     compv,              // compute eigenvectors or not
   matrixf&              Q,                  // orthogonal matrix used in the reduction to tridiagonal form
   vectorf&              eigen_values,       // vector of computed eigenvalues
   matrixf&              eigen_vectors       // matrix of computed eigenvectors
   );

```

Computing for type matrix<complex>

```
bool  matrixc::EigenTridiagonalPosDefQ(
   ENUM_EIGTRIDIAG_Z     compv,              // compute eigenvectors or not
   matrixc&              Q,                  // orthogonal matrix used in the reduction to tridiagonal form
   vector&               eigen_values,       // vector of computed eigenvalues
   matrixc&              eigen_vectors       // matrix of computed eigenvectors
   );

```

Computing for type matrix<complexf>

```
bool  matrixcf::EigenTridiagonalPosDefQ(
   ENUM_EIGTRIDIAG_Z     compv,              // compute eigenvectors or not
   matrixcf&             Q,                  // orthogonal matrix used in the reduction to tridiagonal form
   vectorf&              eigen_values,       // vector of computed eigenvalues
   matrixcf&             eigen_vectors       // matrix of computed eigenvectors
   );

```

Parameters

compv

[in]  ENUM_EIGTRIDIAG_Z enumeration value which determines the method for computing eigenvectors.

Q

[in]  Orthogonal matrix Q produced by method [ReflectTridiagonalToQ](/en/docs/matrix/openblas/matrixtransforms/reflecttridiagonaltoq).

eigen_values

[out] Vector of eigenvalues.

eigen_vectors

[out] Matrix of eigenvectors.

Return Value

Return true if successful, otherwise false in case of an [error](/en/docs/constants/errorswarnings/errorcodes).

Note

Computation depends on the value of the compv parameter.

When compv = EIGCOMPZ_N, compute eigenvalues only, eigenvectors are not calculated.

If EIGCOMPZ_V is set, eigenvalues are computed and eigenvectors of original symmetric matrix are calculated also.

If EIGCOMPZ_I is set, eigenvalues are computed and eigenvectors of tridiagonal matrix are calculated also.

The input must be a symmetric positive definite tridiagonal matrix.

ENUM_EIGTRIDIAG_Z

An enumeration that specifies whether to calculate eigenvectors.

| ID | Description |
| --- | --- |
| EIGCOMPZ_N | 'N': Compute eigenvalues only |
| EIGCOMPZ_V | 'V': Compute eigenvectors of original symmetric matrix also |
| EIGCOMPZ_I | 'I': Compute eigenvectors of tridiagonal matrix also |