claev2 (l) - Linux Manuals

claev2: computes the eigendecomposition of a 2-by-2 Hermitian matrix [ A B ] [ CONJG(B) C ]

NAME

CLAEV2 - computes the eigendecomposition of a 2-by-2 Hermitian matrix [ A B ] [ CONJG(B) C ]

SYNOPSIS

SUBROUTINE CLAEV2(
A, B, C, RT1, RT2, CS1, SN1 )

    
REAL CS1, RT1, RT2

    
COMPLEX A, B, C, SN1

PURPOSE

CLAEV2 computes the eigendecomposition of a 2-by-2 Hermitian matrix
[  A         B  ]
[  CONJG(B)  C  ]. On return, RT1 is the eigenvalue of larger absolute value, RT2 is the eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right eigenvector for RT1, giving the decomposition
[ CS1 CONJG(SN1) ] [ A B ] [ CS1 -CONJG(SN1) ] = [ RT1 0 ] [-SN1 CS1 ] [ CONJG(B) C ] [ SN1 CS1 ] [ 0 RT2 ].

ARGUMENTS

A (input) COMPLEX
The (1,1) element of the 2-by-2 matrix.
B (input) COMPLEX
The (1,2) element and the conjugate of the (2,1) element of the 2-by-2 matrix.
C (input) COMPLEX
The (2,2) element of the 2-by-2 matrix.
RT1 (output) REAL
The eigenvalue of larger absolute value.
RT2 (output) REAL
The eigenvalue of smaller absolute value.
CS1 (output) REAL
SN1 (output) COMPLEX The vector (CS1, SN1) is a unit right eigenvector for RT1.

FURTHER DETAILS

RT1 is accurate to a few ulps barring over/underflow.
RT2 may be inaccurate if there is massive cancellation in the determinant A*C-B*B; higher precision or correctly rounded or correctly truncated arithmetic would be needed to compute RT2 accurately in all cases.
CS1 and SN1 are accurate to a few ulps barring over/underflow. Overflow is possible only if RT1 is within a factor of 5 of overflow. Underflow is harmless if the input data is 0 or exceeds

underflow_threshold / macheps.

Pages related to claev2

  • claev2 (3)
  • claed0 (l) - the divide and conquer method, CLAED0 computes all eigenvalues of a symmetric tridiagonal matrix which is one diagonal block of those from reducing a dense or band Hermitian matrix and corresponding eigenvectors of the dense or band matrix
  • claed7 (l) - computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix
  • claed8 (l) - merges the two sets of eigenvalues together into a single sorted set
  • claein (l) - uses inverse iteration to find a right or left eigenvector corresponding to the eigenvalue W of a complex upper Hessenberg matrix H
  • claesy (l) - computes the eigendecomposition of a 2-by-2 symmetric matrix ( ( A, B );( B, C ) ) provided the norm of the matrix of eigenvectors is larger than some threshold value
  • cla_gbamv (l) - performs one of the matrix-vector operations y := alpha*abs(A)*abs(x) + beta*abs(y),
  • cla_gbrcond_c (l) - CLA_GBRCOND_C Compute the infinity norm condition number of op(A) * inv(diag(C)) where C is a REAL vector