96 lines
4.1 KiB
Markdown
96 lines
4.1 KiB
Markdown
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# EigenTridiagonalDCQ
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Compute eigenvalues and eigenvectors of a symmetric tridiagonal matrix using the divide-and-conquer algorithm (LAPACK function [STEDC](https://www.intel.com/content/www/us/en/docs/onemkl/developer-reference-fortran/2025-2/stedc.html)). Unlike [EigenTridiagonalDC](/en/docs/matrix/openblas/eigen_values/tridiagonalmatrices/eigentridiagposdefq), this method can be used to compute the eigenvectors of the original symmetric matrix. A symmetric matrix can be reduced to tridiagonal form using the [ReduceSymmetricToTridiagonal](/en/docs/matrix/openblas/matrixtransforms/reducesymmetrictotridiagonal) method. The orthogonal matrix Q obtained from this transformation is then used to compute the eigenvectors of the original symmetric matrix.
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Computing for type matrix<double>
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```
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bool matrix::EigenTridiagonalDCQ(
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ENUM_EIGTRIDIAG_Z compv, // compute eigenvectors or not
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matrix& Q, // orthogonal matrix used in the reduction to tridiagonal form
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vector& eigen_values, // vector of computed eigenvalues
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matrix& eigen_vectors // matrix of computed eigenvectors
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);
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```
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Computing for type matrix<float>
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```
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bool matrixf::EigenTridiagonalDCQ(
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ENUM_EIGTRIDIAG_Z compv, // compute eigenvectors or not
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matrixf& Q, // orthogonal matrix used in the reduction to tridiagonal form
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vectorf& eigen_values, // vector of computed eigenvalues
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matrixf& eigen_vectors // matrix of computed eigenvectors
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);
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```
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Computing for type matrix<complex>
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```
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bool matrixc::EigenTridiagonalDCQ(
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ENUM_EIGTRIDIAG_Z compv, // compute eigenvectors or not
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matrixc& Q, // orthogonal matrix used in the reduction to tridiagonal form
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vector& eigen_values, // vector of computed eigenvalues
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matrixc& eigen_vectors // matrix of computed eigenvectors
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);
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```
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Computing for type matrix<complexf>
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```
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bool matrixcf::EigenTridiagonalDCQ(
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ENUM_EIGTRIDIAG_Z compv, // compute eigenvectors or not
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matrixcf& Q, // orthogonal matrix used in the reduction to tridiagonal form
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vectorf& eigen_values, // vector of computed eigenvalues
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matrixcf& eigen_vectors // matrix of computed eigenvectors
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);
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```
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Parameters
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compv
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[in] ENUM_EIGTRIDIAG_Z enumeration value which determines the method for computing eigenvectors.
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Q
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[in] Orthogonal matrix Q produced by method [ReflectTridiagonalToQ](/en/docs/matrix/openblas/matrixtransforms/reflecttridiagonaltoq).
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eigen_values
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[out] Vector of eigenvalues.
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eigen_vectors
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[out] Matrix of eigenvectors.
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Return Value
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Return true if successful, otherwise false in case of an [error](/en/docs/constants/errorswarnings/errorcodes).
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Note
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Computation depends on the value of the compv parameter.
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When compv = EIGCOMPZ_N, compute eigenvalues only, eigenvectors are not calculated.
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If EIGCOMPZ_V is set, eigenvalues are computed and eigenvectors of original symmetric matrix are calculated also.
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If EIGCOMPZ_I is set, eigenvalues are computed and eigenvectors of tridiagonal matrix are calculated also.
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The input must be a symmetric matrix in the tridiagonal form.
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ENUM_EIGTRIDIAG_Z
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An enumeration that specifies whether to calculate eigenvectors.
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| ID | Description |
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| --- | --- |
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| EIGCOMPZ_N | 'N': Compute eigenvalues only |
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| EIGCOMPZ_V | 'V': Compute eigenvectors of original symmetric matrix also |
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| EIGCOMPZ_I | 'I': Compute eigenvectors of tridiagonal matrix also |
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