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Annotation of rpl/lapack/lapack/zlaesy.f, revision 1.8

1.1       bertrand    1:       SUBROUTINE ZLAESY( A, B, C, RT1, RT2, EVSCAL, CS1, SN1 )
                      2: *
                      3: *  -- LAPACK auxiliary routine (version 3.2) --
                      4: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
                      5: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
                      6: *     November 2006
                      7: *
                      8: *     .. Scalar Arguments ..
                      9:       COMPLEX*16         A, B, C, CS1, EVSCAL, RT1, RT2, SN1
                     10: *     ..
                     11: *
                     12: *  Purpose
                     13: *  =======
                     14: *
                     15: *  ZLAESY computes the eigendecomposition of a 2-by-2 symmetric matrix
                     16: *     ( ( A, B );( B, C ) )
                     17: *  provided the norm of the matrix of eigenvectors is larger than
                     18: *  some threshold value.
                     19: *
                     20: *  RT1 is the eigenvalue of larger absolute value, and RT2 of
                     21: *  smaller absolute value.  If the eigenvectors are computed, then
                     22: *  on return ( CS1, SN1 ) is the unit eigenvector for RT1, hence
                     23: *
                     24: *  [  CS1     SN1   ] . [ A  B ] . [ CS1    -SN1   ] = [ RT1  0  ]
                     25: *  [ -SN1     CS1   ]   [ B  C ]   [ SN1     CS1   ]   [  0  RT2 ]
                     26: *
                     27: *  Arguments
                     28: *  =========
                     29: *
                     30: *  A       (input) COMPLEX*16
                     31: *          The ( 1, 1 ) element of input matrix.
                     32: *
                     33: *  B       (input) COMPLEX*16
                     34: *          The ( 1, 2 ) element of input matrix.  The ( 2, 1 ) element
                     35: *          is also given by B, since the 2-by-2 matrix is symmetric.
                     36: *
                     37: *  C       (input) COMPLEX*16
                     38: *          The ( 2, 2 ) element of input matrix.
                     39: *
                     40: *  RT1     (output) COMPLEX*16
                     41: *          The eigenvalue of larger modulus.
                     42: *
                     43: *  RT2     (output) COMPLEX*16
                     44: *          The eigenvalue of smaller modulus.
                     45: *
                     46: *  EVSCAL  (output) COMPLEX*16
                     47: *          The complex value by which the eigenvector matrix was scaled
                     48: *          to make it orthonormal.  If EVSCAL is zero, the eigenvectors
                     49: *          were not computed.  This means one of two things:  the 2-by-2
                     50: *          matrix could not be diagonalized, or the norm of the matrix
                     51: *          of eigenvectors before scaling was larger than the threshold
                     52: *          value THRESH (set below).
                     53: *
                     54: *  CS1     (output) COMPLEX*16
                     55: *  SN1     (output) COMPLEX*16
                     56: *          If EVSCAL .NE. 0,  ( CS1, SN1 ) is the unit right eigenvector
                     57: *          for RT1.
                     58: *
                     59: * =====================================================================
                     60: *
                     61: *     .. Parameters ..
                     62:       DOUBLE PRECISION   ZERO
                     63:       PARAMETER          ( ZERO = 0.0D0 )
                     64:       DOUBLE PRECISION   ONE
                     65:       PARAMETER          ( ONE = 1.0D0 )
                     66:       COMPLEX*16         CONE
                     67:       PARAMETER          ( CONE = ( 1.0D0, 0.0D0 ) )
                     68:       DOUBLE PRECISION   HALF
                     69:       PARAMETER          ( HALF = 0.5D0 )
                     70:       DOUBLE PRECISION   THRESH
                     71:       PARAMETER          ( THRESH = 0.1D0 )
                     72: *     ..
                     73: *     .. Local Scalars ..
                     74:       DOUBLE PRECISION   BABS, EVNORM, TABS, Z
                     75:       COMPLEX*16         S, T, TMP
                     76: *     ..
                     77: *     .. Intrinsic Functions ..
                     78:       INTRINSIC          ABS, MAX, SQRT
                     79: *     ..
                     80: *     .. Executable Statements ..
                     81: *
                     82: *
                     83: *     Special case:  The matrix is actually diagonal.
                     84: *     To avoid divide by zero later, we treat this case separately.
                     85: *
                     86:       IF( ABS( B ).EQ.ZERO ) THEN
                     87:          RT1 = A
                     88:          RT2 = C
                     89:          IF( ABS( RT1 ).LT.ABS( RT2 ) ) THEN
                     90:             TMP = RT1
                     91:             RT1 = RT2
                     92:             RT2 = TMP
                     93:             CS1 = ZERO
                     94:             SN1 = ONE
                     95:          ELSE
                     96:             CS1 = ONE
                     97:             SN1 = ZERO
                     98:          END IF
                     99:       ELSE
                    100: *
                    101: *        Compute the eigenvalues and eigenvectors.
                    102: *        The characteristic equation is
                    103: *           lambda **2 - (A+C) lambda + (A*C - B*B)
                    104: *        and we solve it using the quadratic formula.
                    105: *
                    106:          S = ( A+C )*HALF
                    107:          T = ( A-C )*HALF
                    108: *
                    109: *        Take the square root carefully to avoid over/under flow.
                    110: *
                    111:          BABS = ABS( B )
                    112:          TABS = ABS( T )
                    113:          Z = MAX( BABS, TABS )
                    114:          IF( Z.GT.ZERO )
                    115:      $      T = Z*SQRT( ( T / Z )**2+( B / Z )**2 )
                    116: *
                    117: *        Compute the two eigenvalues.  RT1 and RT2 are exchanged
                    118: *        if necessary so that RT1 will have the greater magnitude.
                    119: *
                    120:          RT1 = S + T
                    121:          RT2 = S - T
                    122:          IF( ABS( RT1 ).LT.ABS( RT2 ) ) THEN
                    123:             TMP = RT1
                    124:             RT1 = RT2
                    125:             RT2 = TMP
                    126:          END IF
                    127: *
                    128: *        Choose CS1 = 1 and SN1 to satisfy the first equation, then
                    129: *        scale the components of this eigenvector so that the matrix
1.8     ! bertrand  130: *        of eigenvectors X satisfies  X * X**T = I .  (No scaling is
1.1       bertrand  131: *        done if the norm of the eigenvalue matrix is less than THRESH.)
                    132: *
                    133:          SN1 = ( RT1-A ) / B
                    134:          TABS = ABS( SN1 )
                    135:          IF( TABS.GT.ONE ) THEN
                    136:             T = TABS*SQRT( ( ONE / TABS )**2+( SN1 / TABS )**2 )
                    137:          ELSE
                    138:             T = SQRT( CONE+SN1*SN1 )
                    139:          END IF
                    140:          EVNORM = ABS( T )
                    141:          IF( EVNORM.GE.THRESH ) THEN
                    142:             EVSCAL = CONE / T
                    143:             CS1 = EVSCAL
                    144:             SN1 = SN1*EVSCAL
                    145:          ELSE
                    146:             EVSCAL = ZERO
                    147:          END IF
                    148:       END IF
                    149:       RETURN
                    150: *
                    151: *     End of ZLAESY
                    152: *
                    153:       END