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annotate libcruft/lapack/slazq4.f @ 8489:e76b92c7f779

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[docs] low cost => low-cost
author Brian Gough <bjg@gnu.org>
date Tue, 13 Jan 2009 00:26:46 -0500
parents 82be108cc558
children
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7789
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1 SUBROUTINE SLAZQ4( I0, N0, Z, PP, N0IN, DMIN, DMIN1, DMIN2, DN,
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2 $ DN1, DN2, TAU, TTYPE, G )
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3 *
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4 * -- LAPACK auxiliary routine (version 3.1) --
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5 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
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6 * November 2006
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7 *
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8 * .. Scalar Arguments ..
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9 INTEGER I0, N0, N0IN, PP, TTYPE
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10 REAL DMIN, DMIN1, DMIN2, DN, DN1, DN2, G, TAU
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11 * ..
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12 * .. Array Arguments ..
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13 REAL Z( * )
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14 * ..
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15 *
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16 * Purpose
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17 * =======
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18 *
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19 * SLAZQ4 computes an approximation TAU to the smallest eigenvalue
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20 * using values of d from the previous transform.
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21 *
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22 * I0 (input) INTEGER
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23 * First index.
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24 *
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25 * N0 (input) INTEGER
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26 * Last index.
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27 *
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28 * Z (input) REAL array, dimension ( 4*N )
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29 * Z holds the qd array.
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30 *
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31 * PP (input) INTEGER
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32 * PP=0 for ping, PP=1 for pong.
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33 *
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34 * N0IN (input) INTEGER
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35 * The value of N0 at start of EIGTEST.
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36 *
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37 * DMIN (input) REAL
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38 * Minimum value of d.
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39 *
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40 * DMIN1 (input) REAL
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41 * Minimum value of d, excluding D( N0 ).
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42 *
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43 * DMIN2 (input) REAL
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44 * Minimum value of d, excluding D( N0 ) and D( N0-1 ).
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45 *
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46 * DN (input) REAL
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47 * d(N)
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48 *
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49 * DN1 (input) REAL
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50 * d(N-1)
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51 *
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52 * DN2 (input) REAL
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53 * d(N-2)
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54 *
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55 * TAU (output) REAL
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56 * This is the shift.
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57 *
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58 * TTYPE (output) INTEGER
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59 * Shift type.
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60 *
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61 * G (input/output) REAL
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62 * G is passed as an argument in order to save its value between
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63 * calls to SLAZQ4
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64 *
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65 * Further Details
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66 * ===============
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67 * CNST1 = 9/16
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68 *
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69 * This is a thread safe version of SLASQ4, which passes G through the
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70 * argument list in place of declaring G in a SAVE statment.
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71 *
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72 * =====================================================================
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73 *
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74 * .. Parameters ..
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75 REAL CNST1, CNST2, CNST3
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76 PARAMETER ( CNST1 = 0.5630E0, CNST2 = 1.010E0,
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77 $ CNST3 = 1.050E0 )
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78 REAL QURTR, THIRD, HALF, ZERO, ONE, TWO, HUNDRD
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79 PARAMETER ( QURTR = 0.250E0, THIRD = 0.3330E0,
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80 $ HALF = 0.50E0, ZERO = 0.0E0, ONE = 1.0E0,
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81 $ TWO = 2.0E0, HUNDRD = 100.0E0 )
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82 * ..
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83 * .. Local Scalars ..
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84 INTEGER I4, NN, NP
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85 REAL A2, B1, B2, GAM, GAP1, GAP2, S
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86 * ..
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87 * .. Intrinsic Functions ..
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88 INTRINSIC MAX, MIN, SQRT
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89 * ..
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90 * .. Executable Statements ..
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91 *
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92 * A negative DMIN forces the shift to take that absolute value
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93 * TTYPE records the type of shift.
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94 *
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95 IF( DMIN.LE.ZERO ) THEN
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96 TAU = -DMIN
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97 TTYPE = -1
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98 RETURN
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99 END IF
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100 *
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101 NN = 4*N0 + PP
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102 IF( N0IN.EQ.N0 ) THEN
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103 *
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104 * No eigenvalues deflated.
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105 *
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106 IF( DMIN.EQ.DN .OR. DMIN.EQ.DN1 ) THEN
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107 *
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108 B1 = SQRT( Z( NN-3 ) )*SQRT( Z( NN-5 ) )
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109 B2 = SQRT( Z( NN-7 ) )*SQRT( Z( NN-9 ) )
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110 A2 = Z( NN-7 ) + Z( NN-5 )
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111 *
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112 * Cases 2 and 3.
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113 *
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114 IF( DMIN.EQ.DN .AND. DMIN1.EQ.DN1 ) THEN
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115 GAP2 = DMIN2 - A2 - DMIN2*QURTR
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116 IF( GAP2.GT.ZERO .AND. GAP2.GT.B2 ) THEN
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117 GAP1 = A2 - DN - ( B2 / GAP2 )*B2
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118 ELSE
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119 GAP1 = A2 - DN - ( B1+B2 )
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120 END IF
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121 IF( GAP1.GT.ZERO .AND. GAP1.GT.B1 ) THEN
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122 S = MAX( DN-( B1 / GAP1 )*B1, HALF*DMIN )
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123 TTYPE = -2
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124 ELSE
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125 S = ZERO
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126 IF( DN.GT.B1 )
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127 $ S = DN - B1
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128 IF( A2.GT.( B1+B2 ) )
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129 $ S = MIN( S, A2-( B1+B2 ) )
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130 S = MAX( S, THIRD*DMIN )
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131 TTYPE = -3
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132 END IF
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133 ELSE
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134 *
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135 * Case 4.
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136 *
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137 TTYPE = -4
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138 S = QURTR*DMIN
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139 IF( DMIN.EQ.DN ) THEN
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140 GAM = DN
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141 A2 = ZERO
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142 IF( Z( NN-5 ) .GT. Z( NN-7 ) )
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143 $ RETURN
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144 B2 = Z( NN-5 ) / Z( NN-7 )
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145 NP = NN - 9
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146 ELSE
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147 NP = NN - 2*PP
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148 B2 = Z( NP-2 )
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149 GAM = DN1
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150 IF( Z( NP-4 ) .GT. Z( NP-2 ) )
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151 $ RETURN
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152 A2 = Z( NP-4 ) / Z( NP-2 )
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153 IF( Z( NN-9 ) .GT. Z( NN-11 ) )
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154 $ RETURN
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155 B2 = Z( NN-9 ) / Z( NN-11 )
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156 NP = NN - 13
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157 END IF
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158 *
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159 * Approximate contribution to norm squared from I < NN-1.
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160 *
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161 A2 = A2 + B2
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162 DO 10 I4 = NP, 4*I0 - 1 + PP, -4
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163 IF( B2.EQ.ZERO )
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164 $ GO TO 20
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165 B1 = B2
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166 IF( Z( I4 ) .GT. Z( I4-2 ) )
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167 $ RETURN
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
168 B2 = B2*( Z( I4 ) / Z( I4-2 ) )
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parents:
diff changeset
169 A2 = A2 + B2
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parents:
diff changeset
170 IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 )
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parents:
diff changeset
171 $ GO TO 20
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parents:
diff changeset
172 10 CONTINUE
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
173 20 CONTINUE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
174 A2 = CNST3*A2
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parents:
diff changeset
175 *
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parents:
diff changeset
176 * Rayleigh quotient residual bound.
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parents:
diff changeset
177 *
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parents:
diff changeset
178 IF( A2.LT.CNST1 )
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parents:
diff changeset
179 $ S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 )
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parents:
diff changeset
180 END IF
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David Bateman <dbateman@free.fr>
parents:
diff changeset
181 ELSE IF( DMIN.EQ.DN2 ) THEN
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parents:
diff changeset
182 *
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parents:
diff changeset
183 * Case 5.
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parents:
diff changeset
184 *
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parents:
diff changeset
185 TTYPE = -5
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parents:
diff changeset
186 S = QURTR*DMIN
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parents:
diff changeset
187 *
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parents:
diff changeset
188 * Compute contribution to norm squared from I > NN-2.
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parents:
diff changeset
189 *
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David Bateman <dbateman@free.fr>
parents:
diff changeset
190 NP = NN - 2*PP
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parents:
diff changeset
191 B1 = Z( NP-2 )
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parents:
diff changeset
192 B2 = Z( NP-6 )
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193 GAM = DN2
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diff changeset
194 IF( Z( NP-8 ).GT.B2 .OR. Z( NP-4 ).GT.B1 )
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parents:
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195 $ RETURN
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196 A2 = ( Z( NP-8 ) / B2 )*( ONE+Z( NP-4 ) / B1 )
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parents:
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197 *
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diff changeset
198 * Approximate contribution to norm squared from I < NN-2.
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parents:
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199 *
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diff changeset
200 IF( N0-I0.GT.2 ) THEN
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parents:
diff changeset
201 B2 = Z( NN-13 ) / Z( NN-15 )
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parents:
diff changeset
202 A2 = A2 + B2
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parents:
diff changeset
203 DO 30 I4 = NN - 17, 4*I0 - 1 + PP, -4
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parents:
diff changeset
204 IF( B2.EQ.ZERO )
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parents:
diff changeset
205 $ GO TO 40
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parents:
diff changeset
206 B1 = B2
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parents:
diff changeset
207 IF( Z( I4 ) .GT. Z( I4-2 ) )
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parents:
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208 $ RETURN
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parents:
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209 B2 = B2*( Z( I4 ) / Z( I4-2 ) )
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parents:
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210 A2 = A2 + B2
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parents:
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211 IF( HUNDRD*MAX( B2, B1 ).LT.A2 .OR. CNST1.LT.A2 )
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parents:
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212 $ GO TO 40
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parents:
diff changeset
213 30 CONTINUE
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parents:
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214 40 CONTINUE
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parents:
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215 A2 = CNST3*A2
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parents:
diff changeset
216 END IF
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parents:
diff changeset
217 *
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parents:
diff changeset
218 IF( A2.LT.CNST1 )
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parents:
diff changeset
219 $ S = GAM*( ONE-SQRT( A2 ) ) / ( ONE+A2 )
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parents:
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220 ELSE
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
221 *
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parents:
diff changeset
222 * Case 6, no information to guide us.
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parents:
diff changeset
223 *
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parents:
diff changeset
224 IF( TTYPE.EQ.-6 ) THEN
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parents:
diff changeset
225 G = G + THIRD*( ONE-G )
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diff changeset
226 ELSE IF( TTYPE.EQ.-18 ) THEN
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parents:
diff changeset
227 G = QURTR*THIRD
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parents:
diff changeset
228 ELSE
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diff changeset
229 G = QURTR
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parents:
diff changeset
230 END IF
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parents:
diff changeset
231 S = G*DMIN
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232 TTYPE = -6
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parents:
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233 END IF
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parents:
diff changeset
234 *
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
235 ELSE IF( N0IN.EQ.( N0+1 ) ) THEN
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parents:
diff changeset
236 *
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diff changeset
237 * One eigenvalue just deflated. Use DMIN1, DN1 for DMIN and DN.
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parents:
diff changeset
238 *
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diff changeset
239 IF( DMIN1.EQ.DN1 .AND. DMIN2.EQ.DN2 ) THEN
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parents:
diff changeset
240 *
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
241 * Cases 7 and 8.
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parents:
diff changeset
242 *
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parents:
diff changeset
243 TTYPE = -7
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244 S = THIRD*DMIN1
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parents:
diff changeset
245 IF( Z( NN-5 ).GT.Z( NN-7 ) )
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parents:
diff changeset
246 $ RETURN
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parents:
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247 B1 = Z( NN-5 ) / Z( NN-7 )
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248 B2 = B1
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diff changeset
249 IF( B2.EQ.ZERO )
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
250 $ GO TO 60
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parents:
diff changeset
251 DO 50 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4
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parents:
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252 A2 = B1
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parents:
diff changeset
253 IF( Z( I4 ).GT.Z( I4-2 ) )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
254 $ RETURN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
255 B1 = B1*( Z( I4 ) / Z( I4-2 ) )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
256 B2 = B2 + B1
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
257 IF( HUNDRD*MAX( B1, A2 ).LT.B2 )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
258 $ GO TO 60
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David Bateman <dbateman@free.fr>
parents:
diff changeset
259 50 CONTINUE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
260 60 CONTINUE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
261 B2 = SQRT( CNST3*B2 )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
262 A2 = DMIN1 / ( ONE+B2**2 )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
263 GAP2 = HALF*DMIN2 - A2
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parents:
diff changeset
264 IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
265 S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) )
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
266 ELSE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
267 S = MAX( S, A2*( ONE-CNST2*B2 ) )
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parents:
diff changeset
268 TTYPE = -8
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parents:
diff changeset
269 END IF
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
270 ELSE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
271 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
272 * Case 9.
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
273 *
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
274 S = QURTR*DMIN1
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
275 IF( DMIN1.EQ.DN1 )
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
276 $ S = HALF*DMIN1
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
277 TTYPE = -9
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parents:
diff changeset
278 END IF
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
279 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
280 ELSE IF( N0IN.EQ.( N0+2 ) ) THEN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
281 *
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
282 * Two eigenvalues deflated. Use DMIN2, DN2 for DMIN and DN.
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
283 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
284 * Cases 10 and 11.
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
285 *
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parents:
diff changeset
286 IF( DMIN2.EQ.DN2 .AND. TWO*Z( NN-5 ).LT.Z( NN-7 ) ) THEN
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
287 TTYPE = -10
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
288 S = THIRD*DMIN2
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
289 IF( Z( NN-5 ).GT.Z( NN-7 ) )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
290 $ RETURN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
291 B1 = Z( NN-5 ) / Z( NN-7 )
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
292 B2 = B1
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
293 IF( B2.EQ.ZERO )
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
294 $ GO TO 80
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
295 DO 70 I4 = 4*N0 - 9 + PP, 4*I0 - 1 + PP, -4
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
296 IF( Z( I4 ).GT.Z( I4-2 ) )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
297 $ RETURN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
298 B1 = B1*( Z( I4 ) / Z( I4-2 ) )
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
299 B2 = B2 + B1
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
300 IF( HUNDRD*B1.LT.B2 )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
301 $ GO TO 80
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
302 70 CONTINUE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
303 80 CONTINUE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
304 B2 = SQRT( CNST3*B2 )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
305 A2 = DMIN2 / ( ONE+B2**2 )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
306 GAP2 = Z( NN-7 ) + Z( NN-9 ) -
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
307 $ SQRT( Z( NN-11 ) )*SQRT( Z( NN-9 ) ) - A2
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parents:
diff changeset
308 IF( GAP2.GT.ZERO .AND. GAP2.GT.B2*A2 ) THEN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
309 S = MAX( S, A2*( ONE-CNST2*A2*( B2 / GAP2 )*B2 ) )
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
310 ELSE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
311 S = MAX( S, A2*( ONE-CNST2*B2 ) )
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
312 END IF
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
313 ELSE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
314 S = QURTR*DMIN2
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
315 TTYPE = -11
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
316 END IF
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
317 ELSE IF( N0IN.GT.( N0+2 ) ) THEN
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
318 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
319 * Case 12, more than two eigenvalues deflated. No information.
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parents:
diff changeset
320 *
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
321 S = ZERO
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
322 TTYPE = -12
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
323 END IF
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
324 *
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
325 TAU = S
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
326 RETURN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
327 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
328 * End of SLAZQ4
82be108cc558 First attempt at single precision tyeps
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parents:
diff changeset
329 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
330 END