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1 SUBROUTINE DSTERF( N, D, E, INFO ) |
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2 * |
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3 * -- LAPACK routine (version 3.0) -- |
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4 * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., |
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5 * Courant Institute, Argonne National Lab, and Rice University |
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6 * June 30, 1999 |
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7 * |
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8 * .. Scalar Arguments .. |
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9 INTEGER INFO, N |
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10 * .. |
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11 * .. Array Arguments .. |
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12 DOUBLE PRECISION D( * ), E( * ) |
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13 * .. |
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14 * |
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15 * Purpose |
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16 * ======= |
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17 * |
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18 * DSTERF computes all eigenvalues of a symmetric tridiagonal matrix |
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19 * using the Pal-Walker-Kahan variant of the QL or QR algorithm. |
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20 * |
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21 * Arguments |
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22 * ========= |
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23 * |
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24 * N (input) INTEGER |
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25 * The order of the matrix. N >= 0. |
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26 * |
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27 * D (input/output) DOUBLE PRECISION array, dimension (N) |
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28 * On entry, the n diagonal elements of the tridiagonal matrix. |
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29 * On exit, if INFO = 0, the eigenvalues in ascending order. |
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30 * |
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31 * E (input/output) DOUBLE PRECISION array, dimension (N-1) |
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32 * On entry, the (n-1) subdiagonal elements of the tridiagonal |
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33 * matrix. |
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34 * On exit, E has been destroyed. |
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35 * |
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36 * INFO (output) INTEGER |
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37 * = 0: successful exit |
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38 * < 0: if INFO = -i, the i-th argument had an illegal value |
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39 * > 0: the algorithm failed to find all of the eigenvalues in |
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40 * a total of 30*N iterations; if INFO = i, then i |
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41 * elements of E have not converged to zero. |
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42 * |
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43 * ===================================================================== |
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44 * |
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45 * .. Parameters .. |
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46 DOUBLE PRECISION ZERO, ONE, TWO, THREE |
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47 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0, |
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48 $ THREE = 3.0D0 ) |
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49 INTEGER MAXIT |
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50 PARAMETER ( MAXIT = 30 ) |
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51 * .. |
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52 * .. Local Scalars .. |
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53 INTEGER I, ISCALE, JTOT, L, L1, LEND, LENDSV, LSV, M, |
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54 $ NMAXIT |
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55 DOUBLE PRECISION ALPHA, ANORM, BB, C, EPS, EPS2, GAMMA, OLDC, |
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56 $ OLDGAM, P, R, RT1, RT2, RTE, S, SAFMAX, SAFMIN, |
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57 $ SIGMA, SSFMAX, SSFMIN |
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58 * .. |
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59 * .. External Functions .. |
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60 DOUBLE PRECISION DLAMCH, DLANST, DLAPY2 |
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61 EXTERNAL DLAMCH, DLANST, DLAPY2 |
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62 * .. |
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63 * .. External Subroutines .. |
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64 EXTERNAL DLAE2, DLASCL, DLASRT, XERBLA |
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65 * .. |
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66 * .. Intrinsic Functions .. |
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67 INTRINSIC ABS, SIGN, SQRT |
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68 * .. |
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69 * .. Executable Statements .. |
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70 * |
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71 * Test the input parameters. |
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72 * |
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73 INFO = 0 |
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74 * |
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75 * Quick return if possible |
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76 * |
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77 IF( N.LT.0 ) THEN |
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78 INFO = -1 |
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79 CALL XERBLA( 'DSTERF', -INFO ) |
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80 RETURN |
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81 END IF |
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82 IF( N.LE.1 ) |
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83 $ RETURN |
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84 * |
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85 * Determine the unit roundoff for this environment. |
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86 * |
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87 EPS = DLAMCH( 'E' ) |
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88 EPS2 = EPS**2 |
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89 SAFMIN = DLAMCH( 'S' ) |
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90 SAFMAX = ONE / SAFMIN |
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91 SSFMAX = SQRT( SAFMAX ) / THREE |
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92 SSFMIN = SQRT( SAFMIN ) / EPS2 |
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93 * |
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94 * Compute the eigenvalues of the tridiagonal matrix. |
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95 * |
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96 NMAXIT = N*MAXIT |
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97 SIGMA = ZERO |
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98 JTOT = 0 |
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99 * |
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100 * Determine where the matrix splits and choose QL or QR iteration |
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101 * for each block, according to whether top or bottom diagonal |
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102 * element is smaller. |
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103 * |
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104 L1 = 1 |
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105 * |
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106 10 CONTINUE |
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107 IF( L1.GT.N ) |
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108 $ GO TO 170 |
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109 IF( L1.GT.1 ) |
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110 $ E( L1-1 ) = ZERO |
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111 DO 20 M = L1, N - 1 |
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112 IF( ABS( E( M ) ).LE.( SQRT( ABS( D( M ) ) )*SQRT( ABS( D( M+ |
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113 $ 1 ) ) ) )*EPS ) THEN |
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114 E( M ) = ZERO |
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115 GO TO 30 |
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116 END IF |
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117 20 CONTINUE |
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118 M = N |
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119 * |
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120 30 CONTINUE |
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121 L = L1 |
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122 LSV = L |
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123 LEND = M |
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124 LENDSV = LEND |
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125 L1 = M + 1 |
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126 IF( LEND.EQ.L ) |
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127 $ GO TO 10 |
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128 * |
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129 * Scale submatrix in rows and columns L to LEND |
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130 * |
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131 ANORM = DLANST( 'I', LEND-L+1, D( L ), E( L ) ) |
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132 ISCALE = 0 |
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133 IF( ANORM.GT.SSFMAX ) THEN |
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134 ISCALE = 1 |
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135 CALL DLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L+1, 1, D( L ), N, |
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136 $ INFO ) |
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137 CALL DLASCL( 'G', 0, 0, ANORM, SSFMAX, LEND-L, 1, E( L ), N, |
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138 $ INFO ) |
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139 ELSE IF( ANORM.LT.SSFMIN ) THEN |
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140 ISCALE = 2 |
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141 CALL DLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L+1, 1, D( L ), N, |
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142 $ INFO ) |
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143 CALL DLASCL( 'G', 0, 0, ANORM, SSFMIN, LEND-L, 1, E( L ), N, |
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144 $ INFO ) |
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145 END IF |
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146 * |
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147 DO 40 I = L, LEND - 1 |
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148 E( I ) = E( I )**2 |
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149 40 CONTINUE |
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150 * |
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151 * Choose between QL and QR iteration |
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152 * |
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153 IF( ABS( D( LEND ) ).LT.ABS( D( L ) ) ) THEN |
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154 LEND = LSV |
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155 L = LENDSV |
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156 END IF |
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157 * |
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158 IF( LEND.GE.L ) THEN |
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159 * |
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160 * QL Iteration |
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161 * |
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162 * Look for small subdiagonal element. |
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163 * |
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164 50 CONTINUE |
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165 IF( L.NE.LEND ) THEN |
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166 DO 60 M = L, LEND - 1 |
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167 IF( ABS( E( M ) ).LE.EPS2*ABS( D( M )*D( M+1 ) ) ) |
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168 $ GO TO 70 |
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169 60 CONTINUE |
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170 END IF |
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171 M = LEND |
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172 * |
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173 70 CONTINUE |
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174 IF( M.LT.LEND ) |
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175 $ E( M ) = ZERO |
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176 P = D( L ) |
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177 IF( M.EQ.L ) |
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178 $ GO TO 90 |
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179 * |
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180 * If remaining matrix is 2 by 2, use DLAE2 to compute its |
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181 * eigenvalues. |
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182 * |
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183 IF( M.EQ.L+1 ) THEN |
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184 RTE = SQRT( E( L ) ) |
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185 CALL DLAE2( D( L ), RTE, D( L+1 ), RT1, RT2 ) |
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186 D( L ) = RT1 |
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187 D( L+1 ) = RT2 |
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188 E( L ) = ZERO |
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189 L = L + 2 |
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190 IF( L.LE.LEND ) |
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191 $ GO TO 50 |
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192 GO TO 150 |
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193 END IF |
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194 * |
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195 IF( JTOT.EQ.NMAXIT ) |
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196 $ GO TO 150 |
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197 JTOT = JTOT + 1 |
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198 * |
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199 * Form shift. |
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200 * |
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201 RTE = SQRT( E( L ) ) |
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202 SIGMA = ( D( L+1 )-P ) / ( TWO*RTE ) |
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203 R = DLAPY2( SIGMA, ONE ) |
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204 SIGMA = P - ( RTE / ( SIGMA+SIGN( R, SIGMA ) ) ) |
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205 * |
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206 C = ONE |
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207 S = ZERO |
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208 GAMMA = D( M ) - SIGMA |
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209 P = GAMMA*GAMMA |
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210 * |
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211 * Inner loop |
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212 * |
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213 DO 80 I = M - 1, L, -1 |
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214 BB = E( I ) |
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215 R = P + BB |
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216 IF( I.NE.M-1 ) |
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217 $ E( I+1 ) = S*R |
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218 OLDC = C |
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219 C = P / R |
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220 S = BB / R |
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221 OLDGAM = GAMMA |
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222 ALPHA = D( I ) |
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223 GAMMA = C*( ALPHA-SIGMA ) - S*OLDGAM |
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224 D( I+1 ) = OLDGAM + ( ALPHA-GAMMA ) |
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225 IF( C.NE.ZERO ) THEN |
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226 P = ( GAMMA*GAMMA ) / C |
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227 ELSE |
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228 P = OLDC*BB |
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229 END IF |
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230 80 CONTINUE |
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231 * |
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232 E( L ) = S*P |
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233 D( L ) = SIGMA + GAMMA |
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234 GO TO 50 |
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235 * |
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236 * Eigenvalue found. |
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237 * |
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238 90 CONTINUE |
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239 D( L ) = P |
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240 * |
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241 L = L + 1 |
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242 IF( L.LE.LEND ) |
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243 $ GO TO 50 |
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244 GO TO 150 |
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245 * |
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246 ELSE |
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247 * |
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248 * QR Iteration |
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249 * |
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250 * Look for small superdiagonal element. |
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251 * |
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252 100 CONTINUE |
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253 DO 110 M = L, LEND + 1, -1 |
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254 IF( ABS( E( M-1 ) ).LE.EPS2*ABS( D( M )*D( M-1 ) ) ) |
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255 $ GO TO 120 |
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256 110 CONTINUE |
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257 M = LEND |
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258 * |
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259 120 CONTINUE |
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260 IF( M.GT.LEND ) |
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261 $ E( M-1 ) = ZERO |
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262 P = D( L ) |
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263 IF( M.EQ.L ) |
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264 $ GO TO 140 |
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265 * |
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266 * If remaining matrix is 2 by 2, use DLAE2 to compute its |
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267 * eigenvalues. |
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268 * |
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269 IF( M.EQ.L-1 ) THEN |
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270 RTE = SQRT( E( L-1 ) ) |
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271 CALL DLAE2( D( L ), RTE, D( L-1 ), RT1, RT2 ) |
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272 D( L ) = RT1 |
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273 D( L-1 ) = RT2 |
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274 E( L-1 ) = ZERO |
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275 L = L - 2 |
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276 IF( L.GE.LEND ) |
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277 $ GO TO 100 |
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278 GO TO 150 |
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279 END IF |
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280 * |
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281 IF( JTOT.EQ.NMAXIT ) |
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282 $ GO TO 150 |
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283 JTOT = JTOT + 1 |
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284 * |
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285 * Form shift. |
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286 * |
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287 RTE = SQRT( E( L-1 ) ) |
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288 SIGMA = ( D( L-1 )-P ) / ( TWO*RTE ) |
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289 R = DLAPY2( SIGMA, ONE ) |
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290 SIGMA = P - ( RTE / ( SIGMA+SIGN( R, SIGMA ) ) ) |
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291 * |
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292 C = ONE |
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293 S = ZERO |
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294 GAMMA = D( M ) - SIGMA |
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295 P = GAMMA*GAMMA |
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296 * |
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297 * Inner loop |
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298 * |
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299 DO 130 I = M, L - 1 |
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300 BB = E( I ) |
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301 R = P + BB |
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302 IF( I.NE.M ) |
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303 $ E( I-1 ) = S*R |
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304 OLDC = C |
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305 C = P / R |
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306 S = BB / R |
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307 OLDGAM = GAMMA |
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308 ALPHA = D( I+1 ) |
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309 GAMMA = C*( ALPHA-SIGMA ) - S*OLDGAM |
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310 D( I ) = OLDGAM + ( ALPHA-GAMMA ) |
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311 IF( C.NE.ZERO ) THEN |
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312 P = ( GAMMA*GAMMA ) / C |
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313 ELSE |
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314 P = OLDC*BB |
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315 END IF |
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316 130 CONTINUE |
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317 * |
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318 E( L-1 ) = S*P |
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319 D( L ) = SIGMA + GAMMA |
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320 GO TO 100 |
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321 * |
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322 * Eigenvalue found. |
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323 * |
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324 140 CONTINUE |
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325 D( L ) = P |
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326 * |
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327 L = L - 1 |
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328 IF( L.GE.LEND ) |
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329 $ GO TO 100 |
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330 GO TO 150 |
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331 * |
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332 END IF |
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333 * |
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334 * Undo scaling if necessary |
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335 * |
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336 150 CONTINUE |
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337 IF( ISCALE.EQ.1 ) |
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338 $ CALL DLASCL( 'G', 0, 0, SSFMAX, ANORM, LENDSV-LSV+1, 1, |
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339 $ D( LSV ), N, INFO ) |
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340 IF( ISCALE.EQ.2 ) |
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341 $ CALL DLASCL( 'G', 0, 0, SSFMIN, ANORM, LENDSV-LSV+1, 1, |
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342 $ D( LSV ), N, INFO ) |
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343 * |
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344 * Check for no convergence to an eigenvalue after a total |
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345 * of N*MAXIT iterations. |
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346 * |
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347 IF( JTOT.LT.NMAXIT ) |
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348 $ GO TO 10 |
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349 DO 160 I = 1, N - 1 |
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350 IF( E( I ).NE.ZERO ) |
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351 $ INFO = INFO + 1 |
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352 160 CONTINUE |
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353 GO TO 180 |
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354 * |
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355 * Sort eigenvalues in increasing order. |
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356 * |
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357 170 CONTINUE |
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358 CALL DLASRT( 'I', N, D, INFO ) |
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359 * |
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360 180 CONTINUE |
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361 RETURN |
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362 * |
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363 * End of DSTERF |
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364 * |
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365 END |
