slaev2 (l) - Linux Manuals

slaev2: computes the eigendecomposition of a 2-by-2 symmetric matrix [ A B ] [ B C ]

NAME

SLAEV2 - computes the eigendecomposition of a 2-by-2 symmetric matrix [ A B ] [ B C ]

SYNOPSIS

SUBROUTINE SLAEV2(
A, B, C, RT1, RT2, CS1, SN1 )

    
REAL A, B, C, CS1, RT1, RT2, SN1

PURPOSE

SLAEV2 computes the eigendecomposition of a 2-by-2 symmetric matrix
[  A   B  ]
[  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  SN1 ] [  A   B  ] [ CS1 -SN1 ]  =  [ RT1  0  ]

[-SN1  CS1 ] [  B   C  ] [ SN1  CS1 ]     [  0  RT2 ].

ARGUMENTS

A (input) REAL
The (1,1) element of the 2-by-2 matrix.
B (input) REAL
The (1,2) element and the conjugate of the (2,1) element of the 2-by-2 matrix.
C (input) REAL
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) REAL 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 slaev2

  • slaev2 (3)
  • slae2 (l) - computes the eigenvalues of a 2-by-2 symmetric matrix [ A B ] [ B C ]
  • slaebz (l) - contains the iteration loops which compute and use the function N(w), which is the count of eigenvalues of a symmetric tridiagonal matrix T less than or equal to its argument w
  • slaed0 (l) - computes all eigenvalues and corresponding eigenvectors of a symmetric tridiagonal matrix using the divide and conquer method
  • slaed1 (l) - computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix
  • slaed2 (l) - merges the two sets of eigenvalues together into a single sorted set
  • slaed3 (l) - finds the roots of the secular equation, as defined by the values in D, W, and RHO, between 1 and K
  • slaed4 (l) - subroutine compute the I-th updated eigenvalue of a symmetric rank-one modification to a diagonal matrix whose elements are given in the array d, and that D(i) < D(j) for i < j and that RHO > 0
  • slaed5 (l) - subroutine compute the I-th eigenvalue of a symmetric rank-one modification of a 2-by-2 diagonal matrix diag( D ) + RHO The diagonal elements in the array D are assumed to satisfy D(i) < D(j) for i < j
  • slaed6 (l) - computes the positive or negative root (closest to the origin) of z(1) z(2) z(3) f(x) = rho + --------- + ---------- + --------- d(1)-x d(2)-x d(3)-x It is assumed that if ORGATI = .true