LAPACK: SRC/slasq4.f Source File

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00001       SUBROUTINE SLASQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN,
00002      $                   DN1, DN2, TAU, TTYPE, G )
00003 *
00004 *  -- LAPACK routine (version 3.2)                                    --
00005 *
00006 *  -- Contributed by Osni Marques of the Lawrence Berkeley National   --
00007 *  -- Laboratory and Beresford Parlett of the Univ. of California at  --
00008 *  -- Berkeley                                                        --
00009 *  -- November 2008                                                   --
00010 *
00011 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00012 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00013 *
00014 *     .. Scalar Arguments ..
00015       INTEGER            I0, N0, N0IN, PP, TTYPE
00016       REAL               DMIN, DMIN1, DMIN2, DN, DN1, DN2, G, TAU
00017 *     ..
00018 *     .. Array Arguments ..
00019       REAL               Z( * )
00020 *     ..
00021 *
00022 *  Purpose
00023 *  =======
00024 *
00025 *  SLASQ4 computes an approximation TAU to the smallest eigenvalue
00026 *  using values of d from the previous transform.
00027 *
00028 *  I0    (input) INTEGER
00029 *        First index.
00030 *
00031 *  N0    (input) INTEGER
00032 *        Last index.
00033 *
00034 *  Z     (input) REAL array, dimension ( 4*N )
00035 *        Z holds the qd array.
00036 *
00037 *  PP    (input) INTEGER
00038 *        PP=0 for ping, PP=1 for pong.
00039 *
00040 *  NOIN  (input) INTEGER
00041 *        The value of N0 at start of EIGTEST.
00042 *
00043 *  DMIN  (input) REAL
00044 *        Minimum value of d.
00045 *
00046 *  DMIN1 (input) REAL
00047 *        Minimum value of d, excluding D( N0 ).
00048 *
00049 *  DMIN2 (input) REAL
00050 *        Minimum value of d, excluding D( N0 ) and D( N0-1 ).
00051 *
00052 *  DN    (input) REAL
00053 *        d(N)
00054 *
00055 *  DN1   (input) REAL
00056 *        d(N-1)
00057 *
00058 *  DN2   (input) REAL
00059 *        d(N-2)
00060 *
00061 *  TAU   (output) REAL
00062 *        This is the shift.
00063 *
00064 *  TTYPE (output) INTEGER
00065 *        Shift type.
00066 *
00067 *  G     (input/output) REAL
00068 *        G is passed as an argument in order to save its value between
00069 *        calls to SLASQ4.
00070 *
00071 *  Further Details
00072 *  ===============
00073 *  CNST1 = 9/16
00074 *
00075 *  =====================================================================
00076 *
00077 *     .. Parameters ..
00078       REAL               CNST1, CNST2, CNST3
00079       PARAMETER          ( CNST1 = 0.5630E0, CNST2 = 1.010E0,
00080      $                   CNST3 = 1.050E0 )
00081       REAL               QURTR, THIRD, HALF, ZERO, ONE, TWO, HUNDRD
00082       PARAMETER          ( QURTR = 0.250E0, THIRD = 0.3330E0,
00083      $                   HALF = 0.50E0, ZERO = 0.0E0, ONE = 1.0E0,
00084      $                   TWO = 2.0E0, HUNDRD = 100.0E0 )
00085 *     ..
00086 *     .. Local Scalars ..
00087       INTEGER            I4, NN, NP
00088       REAL               A2, B1, B2, GAM, GAP1, GAP2, S
00089 *     ..
00090 *     .. Intrinsic Functions ..
00091       INTRINSIC          MAX, MIN, SQRT
00092 *     ..
00093 *     .. Executable Statements ..
00094 *
00095 *     A negative DMIN forces the shift to take that absolute value
00096 *     TTYPE records the type of shift.
00097 *
00098       IF( DMIN.LE.ZERO ) THEN
00099          TAU = -DMIN
00100          TTYPE = -1
00101          RETURN
00102       END IF
00103 *       
00104       NN = 4*N0 + PP
00105       IF( N0IN.EQ.N0 ) THEN
00106 *
00107 *        No eigenvalues deflated.
00108 *
00109          IF( DMIN.EQ.DN .OR. DMIN.EQ.DN1 ) THEN
00110 *
00111             B1 = SQRT( Z( NN-3 ) )*SQRT( Z( NN-5 ) )
00112             B2 = SQRT( Z( NN-7 ) )*SQRT( Z( NN-9 ) )
00113             A2 = Z( NN-7 ) + Z( NN-5 )
00114 *
00115 *           Cases 2 and 3.
00116 *
00117             IF( DMIN.EQ.DN .AND. DMIN1.EQ.DN1 ) THEN
00118                GAP2 = DMIN2 - A2 - DMIN2*QURTR
00119                IF( GAP2.GT.ZERO .AND. GAP2.GT.B2 ) THEN
00120                   GAP1 = A2 - DN - ( B2 / GAP2 )*B2
00121                ELSE
00122                   GAP1 = A2 - DN - ( B1+B2 )
00123                END IF
00124                IF( GAP1.GT.ZERO .AND. GAP1.GT.B1 ) THEN
00125                   S = MAX( DN-( B1 / GAP1 )*B1, HALF*DMIN )
00126                   TTYPE = -2
00127                ELSE
00128                   S = ZERO
00129                   IF( DN.GT.B1 )
00130      $               S = DN - B1
00131                   IF( A2.GT.( B1+B2 ) )
00132      $               S = MIN( S, A2-( B1+B2 ) )
00133                   S = MAX( S, THIRD*DMIN )
00134                   TTYPE = -3
00135                END IF
00136             ELSE
00137 *
00138 *              Case 4.
00139 *
00140                TTYPE = -4
00141                S = QURTR*DMIN
00142                IF( DMIN.EQ.DN ) THEN
00143                   GAM = DN
00144                   A2 = ZERO
00145                   IF( Z( NN-5 ) .GT. Z( NN-7 ) )
00146      $               RETURN
00147                   B2 = Z( NN-5 ) / Z( NN-7 )
00148                   NP = NN - 9
00149                ELSE
00150                   NP = NN - 2*PP
00151                   B2 = Z( NP-2 )
00152                   GAM = DN1
00153                   IF( Z( NP-4 ) .GT. Z( NP-2 ) )
00154      $               RETURN
00155                   A2 = Z( NP-4 ) / Z( NP-2 )
00156                   IF( Z( NN-9 ) .GT. Z( NN-11 ) )
00157      $               RETURN
00158                   B2 = Z( NN-9 ) / Z( NN-11 )
00159                   NP = NN - 13
00160                END IF
00161 *
00162 *              Approximate contribution to norm squared from I < NN-1.
00163 *
00164                A2 = A2 + B2
00165                DO 10 I4 = NP, 4*I0 - 1 + PP, -4
00166                   IF( B2.EQ.ZERO )
00167      $               GO TO 20
00168                   B1 = B2
00169                   IF( Z( I4 ) .GT. Z( I4-2 ) )
00170      $               RETURN
00171                   B2 = B2*( Z( I4 ) / Z( I4-2 ) )
00172                   A2 = A2 + B2
00173                   IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 ) 
00174      $               GO TO 20
00175    10          CONTINUE
00176    20          CONTINUE
00177                A2 = CNST3*A2
00178 *
00179 *              Rayleigh quotient residual bound.
00180 *
00181                IF( A2.LT.CNST1 )
00182      $            S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 )
00183             END IF
00184          ELSE IF( DMIN.EQ.DN2 ) THEN
00185 *
00186 *           Case 5.
00187 *
00188             TTYPE = -5
00189             S = QURTR*DMIN
00190 *
00191 *           Compute contribution to norm squared from I > NN-2.
00192 *
00193             NP = NN - 2*PP
00194             B1 = Z( NP-2 )
00195             B2 = Z( NP-6 )
00196             GAM = DN2
00197             IF( Z( NP-8 ).GT.B2 .OR. Z( NP-4 ).GT.B1 )
00198      $         RETURN
00199             A2 = ( Z( NP-8 ) / B2 )*( ONE+Z( NP-4 ) / B1 )
00200 *
00201 *           Approximate contribution to norm squared from I < NN-2.
00202 *
00203             IF( N0-I0.GT.2 ) THEN
00204                B2 = Z( NN-13 ) / Z( NN-15 )
00205                A2 = A2 + B2
00206                DO 30 I4 = NN - 17, 4*I0 - 1 + PP, -4
00207                   IF( B2.EQ.ZERO )
00208      $               GO TO 40
00209                   B1 = B2
00210                   IF( Z( I4 ) .GT. Z( I4-2 ) )
00211      $               RETURN
00212                   B2 = B2*( Z( I4 ) / Z( I4-2 ) )
00213                   A2 = A2 + B2
00214                   IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 ) 
00215      $               GO TO 40
00216    30          CONTINUE
00217    40          CONTINUE
00218                A2 = CNST3*A2
00219             END IF
00220 *
00221             IF( A2.LT.CNST1 )
00222      $         S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 )
00223          ELSE
00224 *
00225 *           Case 6, no information to guide us.
00226 *
00227             IF( TTYPE.EQ.-6 ) THEN
00228                G = G + THIRD*( ONE-G )
00229             ELSE IF( TTYPE.EQ.-18 ) THEN
00230                G = QURTR*THIRD
00231             ELSE
00232                G = QURTR
00233             END IF
00234             S = G*DMIN
00235             TTYPE = -6
00236          END IF
00237 *
00238       ELSE IF( N0IN.EQ.( N0+1 ) ) THEN
00239 *
00240 *        One eigenvalue just deflated. Use DMIN1, DN1 for DMIN and DN.
00241 *
00242          IF( DMIN1.EQ.DN1 .AND. DMIN2.EQ.DN2 ) THEN 
00243 *
00244 *           Cases 7 and 8.
00245 *
00246             TTYPE = -7
00247             S = THIRD*DMIN1
00248             IF( Z( NN-5 ).GT.Z( NN-7 ) )
00249      $         RETURN
00250             B1 = Z( NN-5 ) / Z( NN-7 )
00251             B2 = B1
00252             IF( B2.EQ.ZERO )
00253      $         GO TO 60
00254             DO 50 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4
00255                A2 = B1
00256                IF( Z( I4 ).GT.Z( I4-2 ) )
00257      $            RETURN
00258                B1 = B1*( Z( I4 ) / Z( I4-2 ) )
00259                B2 = B2 + B1
00260                IF( HUNDRD*MAX( B1, A2 ).LT.B2 ) 
00261      $            GO TO 60
00262    50       CONTINUE
00263    60       CONTINUE
00264             B2 = SQRT( CNST3*B2 )
00265             A2 = DMIN1 / ( ONE+B2**2 )
00266             GAP2 = HALF*DMIN2 - A2
00267             IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN
00268                S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) )
00269             ELSE 
00270                S = MAX( S, A2*( ONE-CNST2*B2 ) )
00271                TTYPE = -8
00272             END IF
00273          ELSE
00274 *
00275 *           Case 9.
00276 *
00277             S = QURTR*DMIN1
00278             IF( DMIN1.EQ.DN1 )
00279      $         S = HALF*DMIN1
00280             TTYPE = -9
00281          END IF
00282 *
00283       ELSE IF( N0IN.EQ.( N0+2 ) ) THEN
00284 *
00285 *        Two eigenvalues deflated. Use DMIN2, DN2 for DMIN and DN.
00286 *
00287 *        Cases 10 and 11.
00288 *
00289          IF( DMIN2.EQ.DN2 .AND. TWO*Z( NN-5 ).LT.Z( NN-7 ) ) THEN 
00290             TTYPE = -10
00291             S = THIRD*DMIN2
00292             IF( Z( NN-5 ).GT.Z( NN-7 ) )
00293      $         RETURN
00294             B1 = Z( NN-5 ) / Z( NN-7 )
00295             B2 = B1
00296             IF( B2.EQ.ZERO )
00297      $         GO TO 80
00298             DO 70 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4
00299                IF( Z( I4 ).GT.Z( I4-2 ) )
00300      $            RETURN
00301                B1 = B1*( Z( I4 ) / Z( I4-2 ) )
00302                B2 = B2 + B1
00303                IF( HUNDRD*B1.LT.B2 )
00304      $            GO TO 80
00305    70       CONTINUE
00306    80       CONTINUE
00307             B2 = SQRT( CNST3*B2 )
00308             A2 = DMIN2 / ( ONE+B2**2 )
00309             GAP2 = Z( NN-7 ) + Z( NN-9 ) -
00310      $             SQRT( Z( NN-11 ) )*SQRT( Z( NN-9 ) ) - A2
00311             IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN
00312                S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) )
00313             ELSE 
00314                S = MAX( S, A2*( ONE-CNST2*B2 ) )
00315             END IF
00316          ELSE
00317             S = QURTR*DMIN2
00318             TTYPE = -11
00319          END IF
00320       ELSE IF( N0IN.GT.( N0+2 ) ) THEN
00321 *
00322 *        Case 12, more than two eigenvalues deflated. No information.
00323 *
00324          S = ZERO 
00325          TTYPE = -12
00326       END IF
00327 *
00328       TAU = S
00329       RETURN
00330 *
00331 *     End of SLASQ4
00332 *
00333       END