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2329
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1 SUBROUTINE DLASY2( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR, |
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2 $ LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO ) |
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3 * |
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4 * -- LAPACK auxiliary routine (version 3.0) -- |
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2329
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5 * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., |
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6 * Courant Institute, Argonne National Lab, and Rice University |
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7 * October 31, 1992 |
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8 * |
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9 * .. Scalar Arguments .. |
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10 LOGICAL LTRANL, LTRANR |
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11 INTEGER INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2 |
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12 DOUBLE PRECISION SCALE, XNORM |
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13 * .. |
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14 * .. Array Arguments .. |
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15 DOUBLE PRECISION B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ), |
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16 $ X( LDX, * ) |
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17 * .. |
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18 * |
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19 * Purpose |
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20 * ======= |
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21 * |
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22 * DLASY2 solves for the N1 by N2 matrix X, 1 <= N1,N2 <= 2, in |
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23 * |
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24 * op(TL)*X + ISGN*X*op(TR) = SCALE*B, |
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25 * |
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26 * where TL is N1 by N1, TR is N2 by N2, B is N1 by N2, and ISGN = 1 or |
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27 * -1. op(T) = T or T', where T' denotes the transpose of T. |
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28 * |
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29 * Arguments |
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30 * ========= |
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31 * |
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32 * LTRANL (input) LOGICAL |
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33 * On entry, LTRANL specifies the op(TL): |
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34 * = .FALSE., op(TL) = TL, |
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35 * = .TRUE., op(TL) = TL'. |
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36 * |
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37 * LTRANR (input) LOGICAL |
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38 * On entry, LTRANR specifies the op(TR): |
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39 * = .FALSE., op(TR) = TR, |
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40 * = .TRUE., op(TR) = TR'. |
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41 * |
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42 * ISGN (input) INTEGER |
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43 * On entry, ISGN specifies the sign of the equation |
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44 * as described before. ISGN may only be 1 or -1. |
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45 * |
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46 * N1 (input) INTEGER |
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47 * On entry, N1 specifies the order of matrix TL. |
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48 * N1 may only be 0, 1 or 2. |
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49 * |
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50 * N2 (input) INTEGER |
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51 * On entry, N2 specifies the order of matrix TR. |
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52 * N2 may only be 0, 1 or 2. |
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53 * |
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54 * TL (input) DOUBLE PRECISION array, dimension (LDTL,2) |
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55 * On entry, TL contains an N1 by N1 matrix. |
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56 * |
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57 * LDTL (input) INTEGER |
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58 * The leading dimension of the matrix TL. LDTL >= max(1,N1). |
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59 * |
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60 * TR (input) DOUBLE PRECISION array, dimension (LDTR,2) |
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61 * On entry, TR contains an N2 by N2 matrix. |
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62 * |
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63 * LDTR (input) INTEGER |
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64 * The leading dimension of the matrix TR. LDTR >= max(1,N2). |
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65 * |
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66 * B (input) DOUBLE PRECISION array, dimension (LDB,2) |
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67 * On entry, the N1 by N2 matrix B contains the right-hand |
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68 * side of the equation. |
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69 * |
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70 * LDB (input) INTEGER |
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71 * The leading dimension of the matrix B. LDB >= max(1,N1). |
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72 * |
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73 * SCALE (output) DOUBLE PRECISION |
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74 * On exit, SCALE contains the scale factor. SCALE is chosen |
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75 * less than or equal to 1 to prevent the solution overflowing. |
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76 * |
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77 * X (output) DOUBLE PRECISION array, dimension (LDX,2) |
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78 * On exit, X contains the N1 by N2 solution. |
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79 * |
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80 * LDX (input) INTEGER |
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81 * The leading dimension of the matrix X. LDX >= max(1,N1). |
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82 * |
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83 * XNORM (output) DOUBLE PRECISION |
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84 * On exit, XNORM is the infinity-norm of the solution. |
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85 * |
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86 * INFO (output) INTEGER |
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87 * On exit, INFO is set to |
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88 * 0: successful exit. |
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89 * 1: TL and TR have too close eigenvalues, so TL or |
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90 * TR is perturbed to get a nonsingular equation. |
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91 * NOTE: In the interests of speed, this routine does not |
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92 * check the inputs for errors. |
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93 * |
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94 * ===================================================================== |
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95 * |
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96 * .. Parameters .. |
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97 DOUBLE PRECISION ZERO, ONE |
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98 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) |
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99 DOUBLE PRECISION TWO, HALF, EIGHT |
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100 PARAMETER ( TWO = 2.0D+0, HALF = 0.5D+0, EIGHT = 8.0D+0 ) |
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101 * .. |
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102 * .. Local Scalars .. |
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103 LOGICAL BSWAP, XSWAP |
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104 INTEGER I, IP, IPIV, IPSV, J, JP, JPSV, K |
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105 DOUBLE PRECISION BET, EPS, GAM, L21, SGN, SMIN, SMLNUM, TAU1, |
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106 $ TEMP, U11, U12, U22, XMAX |
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107 * .. |
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108 * .. Local Arrays .. |
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109 LOGICAL BSWPIV( 4 ), XSWPIV( 4 ) |
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110 INTEGER JPIV( 4 ), LOCL21( 4 ), LOCU12( 4 ), |
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111 $ LOCU22( 4 ) |
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112 DOUBLE PRECISION BTMP( 4 ), T16( 4, 4 ), TMP( 4 ), X2( 2 ) |
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113 * .. |
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114 * .. External Functions .. |
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115 INTEGER IDAMAX |
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116 DOUBLE PRECISION DLAMCH |
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117 EXTERNAL IDAMAX, DLAMCH |
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118 * .. |
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119 * .. External Subroutines .. |
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120 EXTERNAL DCOPY, DSWAP |
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121 * .. |
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122 * .. Intrinsic Functions .. |
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123 INTRINSIC ABS, MAX |
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124 * .. |
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125 * .. Data statements .. |
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126 DATA LOCU12 / 3, 4, 1, 2 / , LOCL21 / 2, 1, 4, 3 / , |
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127 $ LOCU22 / 4, 3, 2, 1 / |
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128 DATA XSWPIV / .FALSE., .FALSE., .TRUE., .TRUE. / |
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129 DATA BSWPIV / .FALSE., .TRUE., .FALSE., .TRUE. / |
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130 * .. |
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131 * .. Executable Statements .. |
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132 * |
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133 * Do not check the input parameters for errors |
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134 * |
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135 INFO = 0 |
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136 * |
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137 * Quick return if possible |
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138 * |
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139 IF( N1.EQ.0 .OR. N2.EQ.0 ) |
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140 $ RETURN |
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141 * |
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142 * Set constants to control overflow |
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143 * |
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144 EPS = DLAMCH( 'P' ) |
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145 SMLNUM = DLAMCH( 'S' ) / EPS |
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146 SGN = ISGN |
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147 * |
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148 K = N1 + N1 + N2 - 2 |
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149 GO TO ( 10, 20, 30, 50 )K |
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150 * |
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151 * 1 by 1: TL11*X + SGN*X*TR11 = B11 |
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152 * |
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153 10 CONTINUE |
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154 TAU1 = TL( 1, 1 ) + SGN*TR( 1, 1 ) |
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155 BET = ABS( TAU1 ) |
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156 IF( BET.LE.SMLNUM ) THEN |
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157 TAU1 = SMLNUM |
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158 BET = SMLNUM |
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159 INFO = 1 |
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160 END IF |
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161 * |
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162 SCALE = ONE |
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163 GAM = ABS( B( 1, 1 ) ) |
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164 IF( SMLNUM*GAM.GT.BET ) |
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165 $ SCALE = ONE / GAM |
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166 * |
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167 X( 1, 1 ) = ( B( 1, 1 )*SCALE ) / TAU1 |
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168 XNORM = ABS( X( 1, 1 ) ) |
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169 RETURN |
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170 * |
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171 * 1 by 2: |
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172 * TL11*[X11 X12] + ISGN*[X11 X12]*op[TR11 TR12] = [B11 B12] |
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173 * [TR21 TR22] |
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174 * |
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175 20 CONTINUE |
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176 * |
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177 SMIN = MAX( EPS*MAX( ABS( TL( 1, 1 ) ), ABS( TR( 1, 1 ) ), |
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178 $ ABS( TR( 1, 2 ) ), ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) ), |
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179 $ SMLNUM ) |
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180 TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 ) |
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181 TMP( 4 ) = TL( 1, 1 ) + SGN*TR( 2, 2 ) |
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182 IF( LTRANR ) THEN |
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183 TMP( 2 ) = SGN*TR( 2, 1 ) |
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184 TMP( 3 ) = SGN*TR( 1, 2 ) |
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185 ELSE |
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186 TMP( 2 ) = SGN*TR( 1, 2 ) |
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187 TMP( 3 ) = SGN*TR( 2, 1 ) |
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188 END IF |
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189 BTMP( 1 ) = B( 1, 1 ) |
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190 BTMP( 2 ) = B( 1, 2 ) |
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191 GO TO 40 |
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192 * |
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193 * 2 by 1: |
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194 * op[TL11 TL12]*[X11] + ISGN* [X11]*TR11 = [B11] |
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195 * [TL21 TL22] [X21] [X21] [B21] |
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196 * |
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197 30 CONTINUE |
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198 SMIN = MAX( EPS*MAX( ABS( TR( 1, 1 ) ), ABS( TL( 1, 1 ) ), |
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199 $ ABS( TL( 1, 2 ) ), ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) ), |
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200 $ SMLNUM ) |
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201 TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 ) |
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202 TMP( 4 ) = TL( 2, 2 ) + SGN*TR( 1, 1 ) |
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203 IF( LTRANL ) THEN |
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204 TMP( 2 ) = TL( 1, 2 ) |
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205 TMP( 3 ) = TL( 2, 1 ) |
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206 ELSE |
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207 TMP( 2 ) = TL( 2, 1 ) |
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208 TMP( 3 ) = TL( 1, 2 ) |
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209 END IF |
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210 BTMP( 1 ) = B( 1, 1 ) |
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211 BTMP( 2 ) = B( 2, 1 ) |
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212 40 CONTINUE |
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213 * |
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214 * Solve 2 by 2 system using complete pivoting. |
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215 * Set pivots less than SMIN to SMIN. |
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216 * |
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217 IPIV = IDAMAX( 4, TMP, 1 ) |
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218 U11 = TMP( IPIV ) |
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219 IF( ABS( U11 ).LE.SMIN ) THEN |
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220 INFO = 1 |
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221 U11 = SMIN |
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222 END IF |
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223 U12 = TMP( LOCU12( IPIV ) ) |
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224 L21 = TMP( LOCL21( IPIV ) ) / U11 |
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225 U22 = TMP( LOCU22( IPIV ) ) - U12*L21 |
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226 XSWAP = XSWPIV( IPIV ) |
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227 BSWAP = BSWPIV( IPIV ) |
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228 IF( ABS( U22 ).LE.SMIN ) THEN |
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229 INFO = 1 |
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230 U22 = SMIN |
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231 END IF |
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232 IF( BSWAP ) THEN |
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233 TEMP = BTMP( 2 ) |
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234 BTMP( 2 ) = BTMP( 1 ) - L21*TEMP |
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235 BTMP( 1 ) = TEMP |
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236 ELSE |
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237 BTMP( 2 ) = BTMP( 2 ) - L21*BTMP( 1 ) |
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238 END IF |
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239 SCALE = ONE |
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240 IF( ( TWO*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( U22 ) .OR. |
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241 $ ( TWO*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( U11 ) ) THEN |
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242 SCALE = HALF / MAX( ABS( BTMP( 1 ) ), ABS( BTMP( 2 ) ) ) |
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243 BTMP( 1 ) = BTMP( 1 )*SCALE |
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244 BTMP( 2 ) = BTMP( 2 )*SCALE |
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245 END IF |
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246 X2( 2 ) = BTMP( 2 ) / U22 |
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247 X2( 1 ) = BTMP( 1 ) / U11 - ( U12 / U11 )*X2( 2 ) |
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248 IF( XSWAP ) THEN |
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249 TEMP = X2( 2 ) |
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250 X2( 2 ) = X2( 1 ) |
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251 X2( 1 ) = TEMP |
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252 END IF |
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253 X( 1, 1 ) = X2( 1 ) |
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254 IF( N1.EQ.1 ) THEN |
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255 X( 1, 2 ) = X2( 2 ) |
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256 XNORM = ABS( X( 1, 1 ) ) + ABS( X( 1, 2 ) ) |
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257 ELSE |
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258 X( 2, 1 ) = X2( 2 ) |
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259 XNORM = MAX( ABS( X( 1, 1 ) ), ABS( X( 2, 1 ) ) ) |
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260 END IF |
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261 RETURN |
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262 * |
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263 * 2 by 2: |
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264 * op[TL11 TL12]*[X11 X12] +ISGN* [X11 X12]*op[TR11 TR12] = [B11 B12] |
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265 * [TL21 TL22] [X21 X22] [X21 X22] [TR21 TR22] [B21 B22] |
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266 * |
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267 * Solve equivalent 4 by 4 system using complete pivoting. |
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268 * Set pivots less than SMIN to SMIN. |
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269 * |
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270 50 CONTINUE |
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271 SMIN = MAX( ABS( TR( 1, 1 ) ), ABS( TR( 1, 2 ) ), |
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272 $ ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) ) |
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273 SMIN = MAX( SMIN, ABS( TL( 1, 1 ) ), ABS( TL( 1, 2 ) ), |
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274 $ ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) ) |
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275 SMIN = MAX( EPS*SMIN, SMLNUM ) |
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276 BTMP( 1 ) = ZERO |
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277 CALL DCOPY( 16, BTMP, 0, T16, 1 ) |
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278 T16( 1, 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 ) |
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279 T16( 2, 2 ) = TL( 2, 2 ) + SGN*TR( 1, 1 ) |
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280 T16( 3, 3 ) = TL( 1, 1 ) + SGN*TR( 2, 2 ) |
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281 T16( 4, 4 ) = TL( 2, 2 ) + SGN*TR( 2, 2 ) |
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282 IF( LTRANL ) THEN |
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283 T16( 1, 2 ) = TL( 2, 1 ) |
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284 T16( 2, 1 ) = TL( 1, 2 ) |
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285 T16( 3, 4 ) = TL( 2, 1 ) |
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286 T16( 4, 3 ) = TL( 1, 2 ) |
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287 ELSE |
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288 T16( 1, 2 ) = TL( 1, 2 ) |
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289 T16( 2, 1 ) = TL( 2, 1 ) |
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290 T16( 3, 4 ) = TL( 1, 2 ) |
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291 T16( 4, 3 ) = TL( 2, 1 ) |
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292 END IF |
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293 IF( LTRANR ) THEN |
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294 T16( 1, 3 ) = SGN*TR( 1, 2 ) |
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295 T16( 2, 4 ) = SGN*TR( 1, 2 ) |
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296 T16( 3, 1 ) = SGN*TR( 2, 1 ) |
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297 T16( 4, 2 ) = SGN*TR( 2, 1 ) |
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298 ELSE |
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299 T16( 1, 3 ) = SGN*TR( 2, 1 ) |
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300 T16( 2, 4 ) = SGN*TR( 2, 1 ) |
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301 T16( 3, 1 ) = SGN*TR( 1, 2 ) |
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302 T16( 4, 2 ) = SGN*TR( 1, 2 ) |
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303 END IF |
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304 BTMP( 1 ) = B( 1, 1 ) |
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305 BTMP( 2 ) = B( 2, 1 ) |
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306 BTMP( 3 ) = B( 1, 2 ) |
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307 BTMP( 4 ) = B( 2, 2 ) |
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308 * |
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309 * Perform elimination |
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310 * |
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311 DO 100 I = 1, 3 |
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312 XMAX = ZERO |
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313 DO 70 IP = I, 4 |
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314 DO 60 JP = I, 4 |
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315 IF( ABS( T16( IP, JP ) ).GE.XMAX ) THEN |
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316 XMAX = ABS( T16( IP, JP ) ) |
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317 IPSV = IP |
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318 JPSV = JP |
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319 END IF |
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320 60 CONTINUE |
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321 70 CONTINUE |
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322 IF( IPSV.NE.I ) THEN |
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323 CALL DSWAP( 4, T16( IPSV, 1 ), 4, T16( I, 1 ), 4 ) |
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324 TEMP = BTMP( I ) |
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325 BTMP( I ) = BTMP( IPSV ) |
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326 BTMP( IPSV ) = TEMP |
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327 END IF |
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328 IF( JPSV.NE.I ) |
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329 $ CALL DSWAP( 4, T16( 1, JPSV ), 1, T16( 1, I ), 1 ) |
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330 JPIV( I ) = JPSV |
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331 IF( ABS( T16( I, I ) ).LT.SMIN ) THEN |
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332 INFO = 1 |
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333 T16( I, I ) = SMIN |
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334 END IF |
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335 DO 90 J = I + 1, 4 |
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336 T16( J, I ) = T16( J, I ) / T16( I, I ) |
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337 BTMP( J ) = BTMP( J ) - T16( J, I )*BTMP( I ) |
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338 DO 80 K = I + 1, 4 |
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339 T16( J, K ) = T16( J, K ) - T16( J, I )*T16( I, K ) |
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340 80 CONTINUE |
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341 90 CONTINUE |
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342 100 CONTINUE |
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343 IF( ABS( T16( 4, 4 ) ).LT.SMIN ) |
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344 $ T16( 4, 4 ) = SMIN |
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345 SCALE = ONE |
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346 IF( ( EIGHT*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( T16( 1, 1 ) ) .OR. |
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347 $ ( EIGHT*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( T16( 2, 2 ) ) .OR. |
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348 $ ( EIGHT*SMLNUM )*ABS( BTMP( 3 ) ).GT.ABS( T16( 3, 3 ) ) .OR. |
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349 $ ( EIGHT*SMLNUM )*ABS( BTMP( 4 ) ).GT.ABS( T16( 4, 4 ) ) ) THEN |
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350 SCALE = ( ONE / EIGHT ) / MAX( ABS( BTMP( 1 ) ), |
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351 $ ABS( BTMP( 2 ) ), ABS( BTMP( 3 ) ), ABS( BTMP( 4 ) ) ) |
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352 BTMP( 1 ) = BTMP( 1 )*SCALE |
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353 BTMP( 2 ) = BTMP( 2 )*SCALE |
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354 BTMP( 3 ) = BTMP( 3 )*SCALE |
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355 BTMP( 4 ) = BTMP( 4 )*SCALE |
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356 END IF |
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357 DO 120 I = 1, 4 |
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358 K = 5 - I |
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359 TEMP = ONE / T16( K, K ) |
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360 TMP( K ) = BTMP( K )*TEMP |
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361 DO 110 J = K + 1, 4 |
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362 TMP( K ) = TMP( K ) - ( TEMP*T16( K, J ) )*TMP( J ) |
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363 110 CONTINUE |
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364 120 CONTINUE |
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365 DO 130 I = 1, 3 |
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366 IF( JPIV( 4-I ).NE.4-I ) THEN |
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367 TEMP = TMP( 4-I ) |
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368 TMP( 4-I ) = TMP( JPIV( 4-I ) ) |
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369 TMP( JPIV( 4-I ) ) = TEMP |
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370 END IF |
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371 130 CONTINUE |
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372 X( 1, 1 ) = TMP( 1 ) |
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373 X( 2, 1 ) = TMP( 2 ) |
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374 X( 1, 2 ) = TMP( 3 ) |
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375 X( 2, 2 ) = TMP( 4 ) |
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376 XNORM = MAX( ABS( TMP( 1 ) )+ABS( TMP( 3 ) ), |
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377 $ ABS( TMP( 2 ) )+ABS( TMP( 4 ) ) ) |
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378 RETURN |
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379 * |
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380 * End of DLASY2 |
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381 * |
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382 END |
