Hermitian functions

Reform the matrix with the eigenvalues modified by a user-defined function. When the user-defined function is real-valued, the result will remain Hermitian, but when the function is complex-valued, the result is best characterized as normal.

When the user-defined function, say \(f\), is analytic, we can say much more about the result: if the eigenvalue decomposition of the Hermitian matrix \(A\) is \(A=Z \Lambda Z^H\), then

\[f(A) = f(Z \Lambda Z^H) = Z f(\Lambda) Z^H.\]

Two important special cases are \(f(\lambda) = \exp(\lambda)\) and \(f(\lambda)=\exp(i \lambda)\), where the former results in a Hermitian matrix and the latter in a normal (in fact, unitary) matrix.

C++ API

void HermitianFunction(UpperOrLower uplo, Matrix<F> &A, std::function<Real(Real)> f)

void HermitianFunction(UpperOrLower uplo, AbstractDistMatrix<F> &A, std::function<Real(Real)> f)

Modifies the eigenvalues of the passed-in Hermitian matrix by replacing each eigenvalue \(\lambda_i\) with \(f(\lambda_i) \in \mathbb{R}\). See examples/lapack_like/RealSymmetricFunction.cpp for an example usage.

void HermitianFunction(UpperOrLower uplo, Matrix<Complex<Real>> &A, std::function<Complex<Real>(Real)> f)

void HermitianFunction(UpperOrLower uplo, AbstractDistMatrix<Complex<Real>> &A, std::function<Complex<Real>(Real)> f)

Modifies the eigenvalues of the passed-in complex Hermitian matrix by replacing each eigenvalue \(\lambda_i\) with \(f(\lambda_i) \in \mathbb{C}\). See examples/lapack_like/ComplexHermitianFunction.cpp for an example usage.

TODO: A version of (complex) HermitianFunction which begins with a real matrix

C API

ElError ElRealHermitianFunction_s(ElUpperOrLower uplo, ElMatrix_s A, float (*func)(float))

ElError ElRealHermitianFunction_d(ElUpperOrLower uplo, ElMatrix_d A, double (*func)(double))

ElError ElRealHermitianFunction_c(ElUpperOrLower uplo, ElMatrix_c A, float (*func)(float))

ElError ElRealHermitianFunction_z(ElUpperOrLower uplo, ElMatrix_z A, double (*func)(double))

ElError ElRealHermitianFunctionDist_s(ElUpperOrLower uplo, ElDistMatrix_s A, float (*func)(float))

ElError ElRealHermitianFunctionDist_d(ElUpperOrLower uplo, ElDistMatrix_d A, double (*func)(double))

ElError ElRealHermitianFunctionDist_c(ElUpperOrLower uplo, ElDistMatrix_c A, float (*func)(float))

ElError ElRealHermitianFunctionDist_z(ElUpperOrLower uplo, ElDistMatrix_z A, double (*func)(double))

Modifies the eigenvalues of the passed-in Hermitian matrix by replacing each eigenvalue \(\lambda_i\) with \(f(\lambda_i) \in \mathbb{R}\).

ElError ElComplexHermitianFunction_c(ElUpperOrLower uplo, ElMatrix_c A, complex_float (*func)(float))

ElError ElComplexHermitianFunction_z(ElUpperOrLower uplo, ElMatrix_z A, complex_double (*func)(double))

ElError ElComplexHermitianFunctionDist_c(ElUpperOrLower uplo, ElDistMatrix_c A, complex_float (*func)(float))

ElError ElComplexHermitianFunctionDist_z(ElUpperOrLower uplo, ElDistMatrix_z A, complex_double (*func)(double))

Modifies the eigenvalues of the passed-in complex Hermitian matrix by replacing each eigenvalue \(\lambda_i\) with \(f(\lambda_i) \in \mathbb{C}\).