107 lines
4.6 KiB
Markdown
107 lines
4.6 KiB
Markdown
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# EigenTridiagonalBisect
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Compute eigenvalues and eigenvectors of a symmetric tridiagonal matrix using the bisection algorithm (LAPACK function [STEVX](https://www.intel.com/content/www/us/en/docs/onemkl/developer-reference-fortran/2025-2/stevx.html)).
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Computing for type matrix<double>
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```
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bool matrix::EigenTridiagonalBisect(
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ENUM_EIG_VALUES jobv, // compute eigenvectors or not
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ENUM_BLAS_RANGE range, // subset of eigenvalues to compute
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double lower, // lower bound of the subset
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double upper, // Upper bound of the subset
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double abstol, // absolute error tolerance
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vector& eigen_values, // vector of computed eigenvectors
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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::EigenTridiagonalBisect(
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ENUM_EIG_VALUES jobv, // compute eigenvectors or not
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ENUM_BLAS_RANGE range, // subset of eigenvalues to compute
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float lower, // lower bound of the subset
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float upper, // upper bound of the subset
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float abstol, // absolute error tolerance
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vectorf& eigen_values, // vector of computed eigenvectors
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matrixf& eigen_vectors // matrix of computed eigenvectors
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);
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```
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Parameters
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jobv
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[in] [ENUM_EIG_VALUES](/en/docs/matrix/openblas/eigen_values/symmetric_matrices/eigensymmetricdc#enum_eig_values) enumeration value which determines the method for computing eigenvectors.
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range
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[in] [ENUM_BLAS_RANGE](/en/docs/matrix/openblas/singular_value_decomposition/singularvaluedecompositionbise) enumeration value that defines a subset of computable eigenvalues and vectors.
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lower
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[in] The lower bound of eigenvalues subset; it is specified depending on the value of the 'range' parameter.
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upper
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[in] The upper bound of eigenvalues subset; it is specified depending on the value of the 'range' parameter.
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abstol
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[in] Absolute error tolerance.
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The absolute error tolerance to which each eigenvalue/eigenvector is required.
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If jobv = 'V', the eigenvalues and eigenvectors output have residual norms bounded by abstol, and the dot products between different eigenvectors are bounded by abstol.
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If abstol < n *eps*|T|, then n *eps*|T| is used instead, where eps is the machine precision, and |T| is the 1-norm of the matrix T. The eigenvalues are computed to an accuracy of eps*|T| irrespective of abstol.
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If high relative precision is important, 'abstol' should be set to a safe minimum value X such that 1.0/X does not overflow.
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eigen_values
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[out] Vector of eigenvalues.
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V
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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 values of the jobv and range parameters.
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When BLASRANGE_A is set, all eigenvalues are computed, and the lower and upper parameters are ignored.
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With the BLASRANGE_V value, only those eigenvalues (and their vectors) are computed, which fall within the range of real values specified by the 'lower' and 'upper' parameters.
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With the BLASRANGE_I value, only those eigenvalues (and their vectors) are computed, which fall within the range of integer indices specified by the 'lower' and 'upper' parameters. For example, with lower=0 and upper=2, only the first three eigenvalues are computed.
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The input must be a symmetric matrix in the tridiagonal form.
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ENUM_EIG_VALUES
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An enumeration defining the need to compute eigenvectors.
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| ID | Description |
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| --- | --- |
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| EIGVALUES_V | Eigenvectors and eigenvalues are calculated. |
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| EIGVALUES_N | Only eigenvalues are computed, without vectors. |
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ENUM_BLAS_RANGE
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An enumeration defining how right singular vectors should be computed.
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| ID | Description |
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| --- | --- |
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| BLASRANGE_A | All singular or eigenvalues will be found. |
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| BLASRANGE_V | All singular or eigenvalues in the half-open interval (VL,VU] will be found. |
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| BLASRANGE_I | The IL-th through IU-th singular or eigenvalues will be found. |
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