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1 SUBROUTINE DLATRS( UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE, |
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2 $ CNORM, INFO ) |
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3 * |
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4 * -- LAPACK auxiliary routine (version 3.0) -- |
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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 * June 30, 1992 |
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8 * |
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9 * .. Scalar Arguments .. |
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10 CHARACTER DIAG, NORMIN, TRANS, UPLO |
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11 INTEGER INFO, LDA, N |
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12 DOUBLE PRECISION SCALE |
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13 * .. |
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14 * .. Array Arguments .. |
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15 DOUBLE PRECISION A( LDA, * ), CNORM( * ), X( * ) |
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16 * .. |
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17 * |
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18 * Purpose |
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19 * ======= |
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20 * |
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21 * DLATRS solves one of the triangular systems |
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22 * |
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23 * A *x = s*b or A'*x = s*b |
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24 * |
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25 * with scaling to prevent overflow. Here A is an upper or lower |
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26 * triangular matrix, A' denotes the transpose of A, x and b are |
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27 * n-element vectors, and s is a scaling factor, usually less than |
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28 * or equal to 1, chosen so that the components of x will be less than |
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29 * the overflow threshold. If the unscaled problem will not cause |
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30 * overflow, the Level 2 BLAS routine DTRSV is called. If the matrix A |
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31 * is singular (A(j,j) = 0 for some j), then s is set to 0 and a |
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32 * non-trivial solution to A*x = 0 is returned. |
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33 * |
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34 * Arguments |
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35 * ========= |
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36 * |
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37 * UPLO (input) CHARACTER*1 |
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38 * Specifies whether the matrix A is upper or lower triangular. |
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39 * = 'U': Upper triangular |
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40 * = 'L': Lower triangular |
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41 * |
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42 * TRANS (input) CHARACTER*1 |
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43 * Specifies the operation applied to A. |
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44 * = 'N': Solve A * x = s*b (No transpose) |
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45 * = 'T': Solve A'* x = s*b (Transpose) |
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46 * = 'C': Solve A'* x = s*b (Conjugate transpose = Transpose) |
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47 * |
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48 * DIAG (input) CHARACTER*1 |
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49 * Specifies whether or not the matrix A is unit triangular. |
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50 * = 'N': Non-unit triangular |
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51 * = 'U': Unit triangular |
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52 * |
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53 * NORMIN (input) CHARACTER*1 |
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54 * Specifies whether CNORM has been set or not. |
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55 * = 'Y': CNORM contains the column norms on entry |
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56 * = 'N': CNORM is not set on entry. On exit, the norms will |
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57 * be computed and stored in CNORM. |
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58 * |
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59 * N (input) INTEGER |
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60 * The order of the matrix A. N >= 0. |
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61 * |
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62 * A (input) DOUBLE PRECISION array, dimension (LDA,N) |
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63 * The triangular matrix A. If UPLO = 'U', the leading n by n |
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64 * upper triangular part of the array A contains the upper |
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65 * triangular matrix, and the strictly lower triangular part of |
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66 * A is not referenced. If UPLO = 'L', the leading n by n lower |
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67 * triangular part of the array A contains the lower triangular |
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68 * matrix, and the strictly upper triangular part of A is not |
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69 * referenced. If DIAG = 'U', the diagonal elements of A are |
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70 * also not referenced and are assumed to be 1. |
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71 * |
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72 * LDA (input) INTEGER |
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73 * The leading dimension of the array A. LDA >= max (1,N). |
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74 * |
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75 * X (input/output) DOUBLE PRECISION array, dimension (N) |
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76 * On entry, the right hand side b of the triangular system. |
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77 * On exit, X is overwritten by the solution vector x. |
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78 * |
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79 * SCALE (output) DOUBLE PRECISION |
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80 * The scaling factor s for the triangular system |
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81 * A * x = s*b or A'* x = s*b. |
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82 * If SCALE = 0, the matrix A is singular or badly scaled, and |
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83 * the vector x is an exact or approximate solution to A*x = 0. |
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84 * |
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85 * CNORM (input or output) DOUBLE PRECISION array, dimension (N) |
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86 * |
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87 * If NORMIN = 'Y', CNORM is an input argument and CNORM(j) |
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88 * contains the norm of the off-diagonal part of the j-th column |
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89 * of A. If TRANS = 'N', CNORM(j) must be greater than or equal |
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90 * to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j) |
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91 * must be greater than or equal to the 1-norm. |
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92 * |
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93 * If NORMIN = 'N', CNORM is an output argument and CNORM(j) |
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94 * returns the 1-norm of the offdiagonal part of the j-th column |
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95 * of A. |
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96 * |
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97 * INFO (output) INTEGER |
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98 * = 0: successful exit |
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99 * < 0: if INFO = -k, the k-th argument had an illegal value |
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100 * |
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101 * Further Details |
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102 * ======= ======= |
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103 * |
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104 * A rough bound on x is computed; if that is less than overflow, DTRSV |
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105 * is called, otherwise, specific code is used which checks for possible |
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106 * overflow or divide-by-zero at every operation. |
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107 * |
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108 * A columnwise scheme is used for solving A*x = b. The basic algorithm |
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109 * if A is lower triangular is |
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110 * |
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111 * x[1:n] := b[1:n] |
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112 * for j = 1, ..., n |
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113 * x(j) := x(j) / A(j,j) |
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114 * x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j] |
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115 * end |
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116 * |
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117 * Define bounds on the components of x after j iterations of the loop: |
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118 * M(j) = bound on x[1:j] |
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119 * G(j) = bound on x[j+1:n] |
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120 * Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}. |
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121 * |
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122 * Then for iteration j+1 we have |
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123 * M(j+1) <= G(j) / | A(j+1,j+1) | |
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124 * G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] | |
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125 * <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | ) |
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126 * |
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127 * where CNORM(j+1) is greater than or equal to the infinity-norm of |
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128 * column j+1 of A, not counting the diagonal. Hence |
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129 * |
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130 * G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | ) |
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131 * 1<=i<=j |
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132 * and |
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133 * |
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134 * |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| ) |
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135 * 1<=i< j |
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136 * |
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137 * Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTRSV if the |
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138 * reciprocal of the largest M(j), j=1,..,n, is larger than |
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139 * max(underflow, 1/overflow). |
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140 * |
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141 * The bound on x(j) is also used to determine when a step in the |
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142 * columnwise method can be performed without fear of overflow. If |
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143 * the computed bound is greater than a large constant, x is scaled to |
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144 * prevent overflow, but if the bound overflows, x is set to 0, x(j) to |
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145 * 1, and scale to 0, and a non-trivial solution to A*x = 0 is found. |
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146 * |
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147 * Similarly, a row-wise scheme is used to solve A'*x = b. The basic |
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148 * algorithm for A upper triangular is |
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149 * |
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150 * for j = 1, ..., n |
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151 * x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j) |
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152 * end |
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153 * |
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154 * We simultaneously compute two bounds |
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155 * G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j |
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156 * M(j) = bound on x(i), 1<=i<=j |
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157 * |
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158 * The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we |
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159 * add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1. |
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160 * Then the bound on x(j) is |
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161 * |
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162 * M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) | |
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163 * |
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164 * <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| ) |
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165 * 1<=i<=j |
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166 * |
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167 * and we can safely call DTRSV if 1/M(n) and 1/G(n) are both greater |
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168 * than max(underflow, 1/overflow). |
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169 * |
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170 * ===================================================================== |
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171 * |
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172 * .. Parameters .. |
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173 DOUBLE PRECISION ZERO, HALF, ONE |
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174 PARAMETER ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0 ) |
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175 * .. |
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176 * .. Local Scalars .. |
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177 LOGICAL NOTRAN, NOUNIT, UPPER |
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178 INTEGER I, IMAX, J, JFIRST, JINC, JLAST |
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179 DOUBLE PRECISION BIGNUM, GROW, REC, SMLNUM, SUMJ, TJJ, TJJS, |
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180 $ TMAX, TSCAL, USCAL, XBND, XJ, XMAX |
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181 * .. |
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182 * .. External Functions .. |
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183 LOGICAL LSAME |
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184 INTEGER IDAMAX |
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185 DOUBLE PRECISION DASUM, DDOT, DLAMCH |
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186 EXTERNAL LSAME, IDAMAX, DASUM, DDOT, DLAMCH |
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187 * .. |
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188 * .. External Subroutines .. |
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189 EXTERNAL DAXPY, DSCAL, DTRSV, XERBLA |
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190 * .. |
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191 * .. Intrinsic Functions .. |
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192 INTRINSIC ABS, MAX, MIN |
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193 * .. |
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194 * .. Executable Statements .. |
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195 * |
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196 INFO = 0 |
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197 UPPER = LSAME( UPLO, 'U' ) |
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198 NOTRAN = LSAME( TRANS, 'N' ) |
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199 NOUNIT = LSAME( DIAG, 'N' ) |
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200 * |
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201 * Test the input parameters. |
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202 * |
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203 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN |
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204 INFO = -1 |
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205 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. |
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206 $ LSAME( TRANS, 'C' ) ) THEN |
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207 INFO = -2 |
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208 ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN |
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209 INFO = -3 |
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210 ELSE IF( .NOT.LSAME( NORMIN, 'Y' ) .AND. .NOT. |
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211 $ LSAME( NORMIN, 'N' ) ) THEN |
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212 INFO = -4 |
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213 ELSE IF( N.LT.0 ) THEN |
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214 INFO = -5 |
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215 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN |
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216 INFO = -7 |
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217 END IF |
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218 IF( INFO.NE.0 ) THEN |
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219 CALL XERBLA( 'DLATRS', -INFO ) |
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220 RETURN |
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221 END IF |
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222 * |
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223 * Quick return if possible |
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224 * |
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225 IF( N.EQ.0 ) |
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226 $ RETURN |
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227 * |
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228 * Determine machine dependent parameters to control overflow. |
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229 * |
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230 SMLNUM = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' ) |
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231 BIGNUM = ONE / SMLNUM |
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232 SCALE = ONE |
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233 * |
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234 IF( LSAME( NORMIN, 'N' ) ) THEN |
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235 * |
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236 * Compute the 1-norm of each column, not including the diagonal. |
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237 * |
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238 IF( UPPER ) THEN |
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239 * |
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240 * A is upper triangular. |
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241 * |
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242 DO 10 J = 1, N |
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243 CNORM( J ) = DASUM( J-1, A( 1, J ), 1 ) |
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244 10 CONTINUE |
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245 ELSE |
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246 * |
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247 * A is lower triangular. |
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248 * |
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249 DO 20 J = 1, N - 1 |
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250 CNORM( J ) = DASUM( N-J, A( J+1, J ), 1 ) |
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251 20 CONTINUE |
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252 CNORM( N ) = ZERO |
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253 END IF |
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254 END IF |
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255 * |
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256 * Scale the column norms by TSCAL if the maximum element in CNORM is |
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257 * greater than BIGNUM. |
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258 * |
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259 IMAX = IDAMAX( N, CNORM, 1 ) |
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260 TMAX = CNORM( IMAX ) |
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261 IF( TMAX.LE.BIGNUM ) THEN |
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262 TSCAL = ONE |
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263 ELSE |
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264 TSCAL = ONE / ( SMLNUM*TMAX ) |
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265 CALL DSCAL( N, TSCAL, CNORM, 1 ) |
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266 END IF |
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267 * |
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268 * Compute a bound on the computed solution vector to see if the |
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269 * Level 2 BLAS routine DTRSV can be used. |
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270 * |
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271 J = IDAMAX( N, X, 1 ) |
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272 XMAX = ABS( X( J ) ) |
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273 XBND = XMAX |
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274 IF( NOTRAN ) THEN |
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275 * |
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276 * Compute the growth in A * x = b. |
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277 * |
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278 IF( UPPER ) THEN |
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279 JFIRST = N |
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280 JLAST = 1 |
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281 JINC = -1 |
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282 ELSE |
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283 JFIRST = 1 |
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284 JLAST = N |
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285 JINC = 1 |
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286 END IF |
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287 * |
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288 IF( TSCAL.NE.ONE ) THEN |
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289 GROW = ZERO |
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290 GO TO 50 |
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291 END IF |
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292 * |
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293 IF( NOUNIT ) THEN |
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294 * |
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295 * A is non-unit triangular. |
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296 * |
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297 * Compute GROW = 1/G(j) and XBND = 1/M(j). |
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298 * Initially, G(0) = max{x(i), i=1,...,n}. |
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299 * |
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300 GROW = ONE / MAX( XBND, SMLNUM ) |
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301 XBND = GROW |
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302 DO 30 J = JFIRST, JLAST, JINC |
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303 * |
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304 * Exit the loop if the growth factor is too small. |
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305 * |
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306 IF( GROW.LE.SMLNUM ) |
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307 $ GO TO 50 |
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308 * |
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309 * M(j) = G(j-1) / abs(A(j,j)) |
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310 * |
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311 TJJ = ABS( A( J, J ) ) |
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312 XBND = MIN( XBND, MIN( ONE, TJJ )*GROW ) |
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313 IF( TJJ+CNORM( J ).GE.SMLNUM ) THEN |
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314 * |
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315 * G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) ) |
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316 * |
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317 GROW = GROW*( TJJ / ( TJJ+CNORM( J ) ) ) |
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318 ELSE |
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319 * |
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320 * G(j) could overflow, set GROW to 0. |
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321 * |
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322 GROW = ZERO |
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323 END IF |
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324 30 CONTINUE |
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325 GROW = XBND |
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326 ELSE |
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327 * |
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328 * A is unit triangular. |
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329 * |
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330 * Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}. |
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331 * |
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332 GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) ) |
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333 DO 40 J = JFIRST, JLAST, JINC |
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334 * |
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335 * Exit the loop if the growth factor is too small. |
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336 * |
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337 IF( GROW.LE.SMLNUM ) |
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338 $ GO TO 50 |
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339 * |
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340 * G(j) = G(j-1)*( 1 + CNORM(j) ) |
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341 * |
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342 GROW = GROW*( ONE / ( ONE+CNORM( J ) ) ) |
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343 40 CONTINUE |
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344 END IF |
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345 50 CONTINUE |
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346 * |
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347 ELSE |
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348 * |
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349 * Compute the growth in A' * x = b. |
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350 * |
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351 IF( UPPER ) THEN |
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352 JFIRST = 1 |
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353 JLAST = N |
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354 JINC = 1 |
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355 ELSE |
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356 JFIRST = N |
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357 JLAST = 1 |
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358 JINC = -1 |
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359 END IF |
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360 * |
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361 IF( TSCAL.NE.ONE ) THEN |
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362 GROW = ZERO |
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363 GO TO 80 |
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364 END IF |
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365 * |
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366 IF( NOUNIT ) THEN |
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367 * |
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368 * A is non-unit triangular. |
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369 * |
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370 * Compute GROW = 1/G(j) and XBND = 1/M(j). |
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371 * Initially, M(0) = max{x(i), i=1,...,n}. |
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372 * |
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373 GROW = ONE / MAX( XBND, SMLNUM ) |
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374 XBND = GROW |
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375 DO 60 J = JFIRST, JLAST, JINC |
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376 * |
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377 * Exit the loop if the growth factor is too small. |
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378 * |
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379 IF( GROW.LE.SMLNUM ) |
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380 $ GO TO 80 |
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381 * |
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382 * G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) ) |
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383 * |
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384 XJ = ONE + CNORM( J ) |
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385 GROW = MIN( GROW, XBND / XJ ) |
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386 * |
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387 * M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j)) |
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388 * |
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389 TJJ = ABS( A( J, J ) ) |
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390 IF( XJ.GT.TJJ ) |
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391 $ XBND = XBND*( TJJ / XJ ) |
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392 60 CONTINUE |
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393 GROW = MIN( GROW, XBND ) |
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394 ELSE |
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395 * |
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396 * A is unit triangular. |
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397 * |
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398 * Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}. |
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399 * |
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400 GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) ) |
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401 DO 70 J = JFIRST, JLAST, JINC |
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402 * |
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403 * Exit the loop if the growth factor is too small. |
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404 * |
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405 IF( GROW.LE.SMLNUM ) |
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406 $ GO TO 80 |
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407 * |
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408 * G(j) = ( 1 + CNORM(j) )*G(j-1) |
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409 * |
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410 XJ = ONE + CNORM( J ) |
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411 GROW = GROW / XJ |
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412 70 CONTINUE |
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413 END IF |
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414 80 CONTINUE |
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415 END IF |
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416 * |
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417 IF( ( GROW*TSCAL ).GT.SMLNUM ) THEN |
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418 * |
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419 * Use the Level 2 BLAS solve if the reciprocal of the bound on |
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420 * elements of X is not too small. |
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421 * |
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422 CALL DTRSV( UPLO, TRANS, DIAG, N, A, LDA, X, 1 ) |
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423 ELSE |
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424 * |
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425 * Use a Level 1 BLAS solve, scaling intermediate results. |
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426 * |
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427 IF( XMAX.GT.BIGNUM ) THEN |
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428 * |
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429 * Scale X so that its components are less than or equal to |
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430 * BIGNUM in absolute value. |
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431 * |
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432 SCALE = BIGNUM / XMAX |
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433 CALL DSCAL( N, SCALE, X, 1 ) |
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434 XMAX = BIGNUM |
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435 END IF |
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436 * |
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437 IF( NOTRAN ) THEN |
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438 * |
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439 * Solve A * x = b |
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440 * |
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441 DO 110 J = JFIRST, JLAST, JINC |
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442 * |
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443 * Compute x(j) = b(j) / A(j,j), scaling x if necessary. |
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444 * |
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445 XJ = ABS( X( J ) ) |
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446 IF( NOUNIT ) THEN |
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447 TJJS = A( J, J )*TSCAL |
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448 ELSE |
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449 TJJS = TSCAL |
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450 IF( TSCAL.EQ.ONE ) |
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451 $ GO TO 100 |
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452 END IF |
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453 TJJ = ABS( TJJS ) |
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454 IF( TJJ.GT.SMLNUM ) THEN |
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455 * |
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456 * abs(A(j,j)) > SMLNUM: |
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457 * |
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458 IF( TJJ.LT.ONE ) THEN |
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459 IF( XJ.GT.TJJ*BIGNUM ) THEN |
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460 * |
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461 * Scale x by 1/b(j). |
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462 * |
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463 REC = ONE / XJ |
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464 CALL DSCAL( N, REC, X, 1 ) |
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465 SCALE = SCALE*REC |
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466 XMAX = XMAX*REC |
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467 END IF |
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468 END IF |
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469 X( J ) = X( J ) / TJJS |
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470 XJ = ABS( X( J ) ) |
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471 ELSE IF( TJJ.GT.ZERO ) THEN |
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472 * |
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473 * 0 < abs(A(j,j)) <= SMLNUM: |
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474 * |
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475 IF( XJ.GT.TJJ*BIGNUM ) THEN |
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476 * |
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477 * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM |
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478 * to avoid overflow when dividing by A(j,j). |
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479 * |
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480 REC = ( TJJ*BIGNUM ) / XJ |
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481 IF( CNORM( J ).GT.ONE ) THEN |
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482 * |
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483 * Scale by 1/CNORM(j) to avoid overflow when |
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484 * multiplying x(j) times column j. |
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485 * |
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486 REC = REC / CNORM( J ) |
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487 END IF |
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488 CALL DSCAL( N, REC, X, 1 ) |
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489 SCALE = SCALE*REC |
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490 XMAX = XMAX*REC |
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491 END IF |
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492 X( J ) = X( J ) / TJJS |
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493 XJ = ABS( X( J ) ) |
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494 ELSE |
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495 * |
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496 * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and |
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497 * scale = 0, and compute a solution to A*x = 0. |
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498 * |
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499 DO 90 I = 1, N |
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500 X( I ) = ZERO |
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501 90 CONTINUE |
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502 X( J ) = ONE |
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503 XJ = ONE |
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504 SCALE = ZERO |
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505 XMAX = ZERO |
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506 END IF |
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507 100 CONTINUE |
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508 * |
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509 * Scale x if necessary to avoid overflow when adding a |
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510 * multiple of column j of A. |
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511 * |
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512 IF( XJ.GT.ONE ) THEN |
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513 REC = ONE / XJ |
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514 IF( CNORM( J ).GT.( BIGNUM-XMAX )*REC ) THEN |
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515 * |
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|
516 * Scale x by 1/(2*abs(x(j))). |
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|
517 * |
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518 REC = REC*HALF |
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519 CALL DSCAL( N, REC, X, 1 ) |
|
|
520 SCALE = SCALE*REC |
|
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521 END IF |
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522 ELSE IF( XJ*CNORM( J ).GT.( BIGNUM-XMAX ) ) THEN |
|
|
523 * |
|
|
524 * Scale x by 1/2. |
|
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525 * |
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526 CALL DSCAL( N, HALF, X, 1 ) |
|
|
527 SCALE = SCALE*HALF |
|
|
528 END IF |
|
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529 * |
|
|
530 IF( UPPER ) THEN |
|
|
531 IF( J.GT.1 ) THEN |
|
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532 * |
|
|
533 * Compute the update |
|
|
534 * x(1:j-1) := x(1:j-1) - x(j) * A(1:j-1,j) |
|
|
535 * |
|
|
536 CALL DAXPY( J-1, -X( J )*TSCAL, A( 1, J ), 1, X, |
|
|
537 $ 1 ) |
|
|
538 I = IDAMAX( J-1, X, 1 ) |
|
|
539 XMAX = ABS( X( I ) ) |
|
|
540 END IF |
|
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541 ELSE |
|
|
542 IF( J.LT.N ) THEN |
|
|
543 * |
|
|
544 * Compute the update |
|
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545 * x(j+1:n) := x(j+1:n) - x(j) * A(j+1:n,j) |
|
|
546 * |
|
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547 CALL DAXPY( N-J, -X( J )*TSCAL, A( J+1, J ), 1, |
|
|
548 $ X( J+1 ), 1 ) |
|
|
549 I = J + IDAMAX( N-J, X( J+1 ), 1 ) |
|
|
550 XMAX = ABS( X( I ) ) |
|
|
551 END IF |
|
|
552 END IF |
|
|
553 110 CONTINUE |
|
|
554 * |
|
|
555 ELSE |
|
|
556 * |
|
|
557 * Solve A' * x = b |
|
|
558 * |
|
|
559 DO 160 J = JFIRST, JLAST, JINC |
|
|
560 * |
|
|
561 * Compute x(j) = b(j) - sum A(k,j)*x(k). |
|
|
562 * k<>j |
|
|
563 * |
|
|
564 XJ = ABS( X( J ) ) |
|
|
565 USCAL = TSCAL |
|
|
566 REC = ONE / MAX( XMAX, ONE ) |
|
|
567 IF( CNORM( J ).GT.( BIGNUM-XJ )*REC ) THEN |
|
|
568 * |
|
|
569 * If x(j) could overflow, scale x by 1/(2*XMAX). |
|
|
570 * |
|
|
571 REC = REC*HALF |
|
|
572 IF( NOUNIT ) THEN |
|
|
573 TJJS = A( J, J )*TSCAL |
|
|
574 ELSE |
|
|
575 TJJS = TSCAL |
|
|
576 END IF |
|
|
577 TJJ = ABS( TJJS ) |
|
|
578 IF( TJJ.GT.ONE ) THEN |
|
|
579 * |
|
|
580 * Divide by A(j,j) when scaling x if A(j,j) > 1. |
|
|
581 * |
|
|
582 REC = MIN( ONE, REC*TJJ ) |
|
|
583 USCAL = USCAL / TJJS |
|
|
584 END IF |
|
|
585 IF( REC.LT.ONE ) THEN |
|
|
586 CALL DSCAL( N, REC, X, 1 ) |
|
|
587 SCALE = SCALE*REC |
|
|
588 XMAX = XMAX*REC |
|
|
589 END IF |
|
|
590 END IF |
|
|
591 * |
|
|
592 SUMJ = ZERO |
|
|
593 IF( USCAL.EQ.ONE ) THEN |
|
|
594 * |
|
|
595 * If the scaling needed for A in the dot product is 1, |
|
|
596 * call DDOT to perform the dot product. |
|
|
597 * |
|
|
598 IF( UPPER ) THEN |
|
|
599 SUMJ = DDOT( J-1, A( 1, J ), 1, X, 1 ) |
|
|
600 ELSE IF( J.LT.N ) THEN |
|
|
601 SUMJ = DDOT( N-J, A( J+1, J ), 1, X( J+1 ), 1 ) |
|
|
602 END IF |
|
|
603 ELSE |
|
|
604 * |
|
|
605 * Otherwise, use in-line code for the dot product. |
|
|
606 * |
|
|
607 IF( UPPER ) THEN |
|
|
608 DO 120 I = 1, J - 1 |
|
|
609 SUMJ = SUMJ + ( A( I, J )*USCAL )*X( I ) |
|
|
610 120 CONTINUE |
|
|
611 ELSE IF( J.LT.N ) THEN |
|
|
612 DO 130 I = J + 1, N |
|
|
613 SUMJ = SUMJ + ( A( I, J )*USCAL )*X( I ) |
|
|
614 130 CONTINUE |
|
|
615 END IF |
|
|
616 END IF |
|
|
617 * |
|
|
618 IF( USCAL.EQ.TSCAL ) THEN |
|
|
619 * |
|
|
620 * Compute x(j) := ( x(j) - sumj ) / A(j,j) if 1/A(j,j) |
|
|
621 * was not used to scale the dotproduct. |
|
|
622 * |
|
|
623 X( J ) = X( J ) - SUMJ |
|
|
624 XJ = ABS( X( J ) ) |
|
|
625 IF( NOUNIT ) THEN |
|
|
626 TJJS = A( J, J )*TSCAL |
|
|
627 ELSE |
|
|
628 TJJS = TSCAL |
|
|
629 IF( TSCAL.EQ.ONE ) |
|
|
630 $ GO TO 150 |
|
|
631 END IF |
|
|
632 * |
|
|
633 * Compute x(j) = x(j) / A(j,j), scaling if necessary. |
|
|
634 * |
|
|
635 TJJ = ABS( TJJS ) |
|
|
636 IF( TJJ.GT.SMLNUM ) THEN |
|
|
637 * |
|
|
638 * abs(A(j,j)) > SMLNUM: |
|
|
639 * |
|
|
640 IF( TJJ.LT.ONE ) THEN |
|
|
641 IF( XJ.GT.TJJ*BIGNUM ) THEN |
|
|
642 * |
|
|
643 * Scale X by 1/abs(x(j)). |
|
|
644 * |
|
|
645 REC = ONE / XJ |
|
|
646 CALL DSCAL( N, REC, X, 1 ) |
|
|
647 SCALE = SCALE*REC |
|
|
648 XMAX = XMAX*REC |
|
|
649 END IF |
|
|
650 END IF |
|
|
651 X( J ) = X( J ) / TJJS |
|
|
652 ELSE IF( TJJ.GT.ZERO ) THEN |
|
|
653 * |
|
|
654 * 0 < abs(A(j,j)) <= SMLNUM: |
|
|
655 * |
|
|
656 IF( XJ.GT.TJJ*BIGNUM ) THEN |
|
|
657 * |
|
|
658 * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM. |
|
|
659 * |
|
|
660 REC = ( TJJ*BIGNUM ) / XJ |
|
|
661 CALL DSCAL( N, REC, X, 1 ) |
|
|
662 SCALE = SCALE*REC |
|
|
663 XMAX = XMAX*REC |
|
|
664 END IF |
|
|
665 X( J ) = X( J ) / TJJS |
|
|
666 ELSE |
|
|
667 * |
|
|
668 * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and |
|
|
669 * scale = 0, and compute a solution to A'*x = 0. |
|
|
670 * |
|
|
671 DO 140 I = 1, N |
|
|
672 X( I ) = ZERO |
|
|
673 140 CONTINUE |
|
|
674 X( J ) = ONE |
|
|
675 SCALE = ZERO |
|
|
676 XMAX = ZERO |
|
|
677 END IF |
|
|
678 150 CONTINUE |
|
|
679 ELSE |
|
|
680 * |
|
|
681 * Compute x(j) := x(j) / A(j,j) - sumj if the dot |
|
|
682 * product has already been divided by 1/A(j,j). |
|
|
683 * |
|
|
684 X( J ) = X( J ) / TJJS - SUMJ |
|
|
685 END IF |
|
|
686 XMAX = MAX( XMAX, ABS( X( J ) ) ) |
|
|
687 160 CONTINUE |
|
|
688 END IF |
|
|
689 SCALE = SCALE / TSCAL |
|
|
690 END IF |
|
|
691 * |
|
|
692 * Scale the column norms by 1/TSCAL for return. |
|
|
693 * |
|
|
694 IF( TSCAL.NE.ONE ) THEN |
|
|
695 CALL DSCAL( N, ONE / TSCAL, CNORM, 1 ) |
|
|
696 END IF |
|
|
697 * |
|
|
698 RETURN |
|
|
699 * |
|
|
700 * End of DLATRS |
|
|
701 * |
|
|
702 END |
