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1 SUBROUTINE DLATBS( UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X, |
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2 $ SCALE, CNORM, INFO ) |
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3 * |
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7034
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4 * -- LAPACK auxiliary routine (version 3.1) -- |
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5 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. |
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6 * November 2006 |
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7 * |
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8 * .. Scalar Arguments .. |
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9 CHARACTER DIAG, NORMIN, TRANS, UPLO |
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10 INTEGER INFO, KD, LDAB, N |
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11 DOUBLE PRECISION SCALE |
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12 * .. |
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13 * .. Array Arguments .. |
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14 DOUBLE PRECISION AB( LDAB, * ), CNORM( * ), X( * ) |
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15 * .. |
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16 * |
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17 * Purpose |
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18 * ======= |
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19 * |
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20 * DLATBS solves one of the triangular systems |
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21 * |
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22 * A *x = s*b or A'*x = s*b |
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23 * |
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24 * with scaling to prevent overflow, where A is an upper or lower |
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25 * triangular band matrix. Here A' denotes the transpose of A, x and b |
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26 * are n-element vectors, and s is a scaling factor, usually less than |
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27 * or equal to 1, chosen so that the components of x will be less than |
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28 * the overflow threshold. If the unscaled problem will not cause |
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29 * overflow, the Level 2 BLAS routine DTBSV is called. If the matrix A |
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30 * is singular (A(j,j) = 0 for some j), then s is set to 0 and a |
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31 * non-trivial solution to A*x = 0 is returned. |
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32 * |
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33 * Arguments |
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34 * ========= |
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35 * |
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36 * UPLO (input) CHARACTER*1 |
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37 * Specifies whether the matrix A is upper or lower triangular. |
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38 * = 'U': Upper triangular |
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39 * = 'L': Lower triangular |
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40 * |
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41 * TRANS (input) CHARACTER*1 |
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42 * Specifies the operation applied to A. |
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43 * = 'N': Solve A * x = s*b (No transpose) |
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44 * = 'T': Solve A'* x = s*b (Transpose) |
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45 * = 'C': Solve A'* x = s*b (Conjugate transpose = Transpose) |
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46 * |
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47 * DIAG (input) CHARACTER*1 |
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48 * Specifies whether or not the matrix A is unit triangular. |
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49 * = 'N': Non-unit triangular |
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50 * = 'U': Unit triangular |
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51 * |
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52 * NORMIN (input) CHARACTER*1 |
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53 * Specifies whether CNORM has been set or not. |
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54 * = 'Y': CNORM contains the column norms on entry |
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55 * = 'N': CNORM is not set on entry. On exit, the norms will |
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56 * be computed and stored in CNORM. |
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57 * |
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58 * N (input) INTEGER |
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59 * The order of the matrix A. N >= 0. |
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60 * |
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61 * KD (input) INTEGER |
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62 * The number of subdiagonals or superdiagonals in the |
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63 * triangular matrix A. KD >= 0. |
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64 * |
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65 * AB (input) DOUBLE PRECISION array, dimension (LDAB,N) |
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66 * The upper or lower triangular band matrix A, stored in the |
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67 * first KD+1 rows of the array. The j-th column of A is stored |
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68 * in the j-th column of the array AB as follows: |
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69 * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; |
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70 * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). |
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71 * |
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72 * LDAB (input) INTEGER |
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73 * The leading dimension of the array AB. LDAB >= KD+1. |
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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, DTBSV |
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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 DTBSV 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 DTBSV 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, JLEN, MAIND |
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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, DTBSV, 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( KD.LT.0 ) THEN |
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216 INFO = -6 |
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217 ELSE IF( LDAB.LT.KD+1 ) THEN |
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218 INFO = -8 |
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219 END IF |
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220 IF( INFO.NE.0 ) THEN |
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221 CALL XERBLA( 'DLATBS', -INFO ) |
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222 RETURN |
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223 END IF |
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224 * |
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225 * Quick return if possible |
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226 * |
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227 IF( N.EQ.0 ) |
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228 $ RETURN |
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229 * |
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230 * Determine machine dependent parameters to control overflow. |
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231 * |
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232 SMLNUM = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' ) |
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233 BIGNUM = ONE / SMLNUM |
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234 SCALE = ONE |
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235 * |
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236 IF( LSAME( NORMIN, 'N' ) ) THEN |
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237 * |
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238 * Compute the 1-norm of each column, not including the diagonal. |
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239 * |
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240 IF( UPPER ) THEN |
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241 * |
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242 * A is upper triangular. |
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243 * |
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244 DO 10 J = 1, N |
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245 JLEN = MIN( KD, J-1 ) |
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246 CNORM( J ) = DASUM( JLEN, AB( KD+1-JLEN, J ), 1 ) |
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247 10 CONTINUE |
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248 ELSE |
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249 * |
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250 * A is lower triangular. |
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251 * |
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252 DO 20 J = 1, N |
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253 JLEN = MIN( KD, N-J ) |
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254 IF( JLEN.GT.0 ) THEN |
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255 CNORM( J ) = DASUM( JLEN, AB( 2, J ), 1 ) |
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256 ELSE |
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257 CNORM( J ) = ZERO |
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258 END IF |
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259 20 CONTINUE |
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260 END IF |
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261 END IF |
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262 * |
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263 * Scale the column norms by TSCAL if the maximum element in CNORM is |
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264 * greater than BIGNUM. |
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265 * |
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266 IMAX = IDAMAX( N, CNORM, 1 ) |
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267 TMAX = CNORM( IMAX ) |
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268 IF( TMAX.LE.BIGNUM ) THEN |
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269 TSCAL = ONE |
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270 ELSE |
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271 TSCAL = ONE / ( SMLNUM*TMAX ) |
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272 CALL DSCAL( N, TSCAL, CNORM, 1 ) |
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273 END IF |
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274 * |
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275 * Compute a bound on the computed solution vector to see if the |
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276 * Level 2 BLAS routine DTBSV can be used. |
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277 * |
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278 J = IDAMAX( N, X, 1 ) |
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279 XMAX = ABS( X( J ) ) |
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280 XBND = XMAX |
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281 IF( NOTRAN ) THEN |
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282 * |
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283 * Compute the growth in A * x = b. |
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284 * |
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285 IF( UPPER ) THEN |
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286 JFIRST = N |
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287 JLAST = 1 |
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288 JINC = -1 |
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289 MAIND = KD + 1 |
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290 ELSE |
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291 JFIRST = 1 |
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292 JLAST = N |
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293 JINC = 1 |
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294 MAIND = 1 |
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295 END IF |
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296 * |
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297 IF( TSCAL.NE.ONE ) THEN |
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298 GROW = ZERO |
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299 GO TO 50 |
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300 END IF |
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301 * |
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302 IF( NOUNIT ) THEN |
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303 * |
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304 * A is non-unit triangular. |
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305 * |
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306 * Compute GROW = 1/G(j) and XBND = 1/M(j). |
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307 * Initially, G(0) = max{x(i), i=1,...,n}. |
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308 * |
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309 GROW = ONE / MAX( XBND, SMLNUM ) |
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310 XBND = GROW |
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311 DO 30 J = JFIRST, JLAST, JINC |
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312 * |
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313 * Exit the loop if the growth factor is too small. |
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314 * |
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315 IF( GROW.LE.SMLNUM ) |
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316 $ GO TO 50 |
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317 * |
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318 * M(j) = G(j-1) / abs(A(j,j)) |
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319 * |
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320 TJJ = ABS( AB( MAIND, J ) ) |
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321 XBND = MIN( XBND, MIN( ONE, TJJ )*GROW ) |
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322 IF( TJJ+CNORM( J ).GE.SMLNUM ) THEN |
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323 * |
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324 * G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) ) |
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325 * |
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326 GROW = GROW*( TJJ / ( TJJ+CNORM( J ) ) ) |
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327 ELSE |
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328 * |
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329 * G(j) could overflow, set GROW to 0. |
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330 * |
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331 GROW = ZERO |
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332 END IF |
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333 30 CONTINUE |
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334 GROW = XBND |
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335 ELSE |
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336 * |
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337 * A is unit triangular. |
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338 * |
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339 * Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}. |
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340 * |
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341 GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) ) |
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342 DO 40 J = JFIRST, JLAST, JINC |
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343 * |
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344 * Exit the loop if the growth factor is too small. |
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345 * |
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346 IF( GROW.LE.SMLNUM ) |
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347 $ GO TO 50 |
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348 * |
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349 * G(j) = G(j-1)*( 1 + CNORM(j) ) |
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350 * |
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351 GROW = GROW*( ONE / ( ONE+CNORM( J ) ) ) |
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352 40 CONTINUE |
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353 END IF |
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354 50 CONTINUE |
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355 * |
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356 ELSE |
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357 * |
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358 * Compute the growth in A' * x = b. |
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359 * |
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360 IF( UPPER ) THEN |
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361 JFIRST = 1 |
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362 JLAST = N |
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363 JINC = 1 |
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364 MAIND = KD + 1 |
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365 ELSE |
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366 JFIRST = N |
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367 JLAST = 1 |
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368 JINC = -1 |
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369 MAIND = 1 |
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370 END IF |
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371 * |
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372 IF( TSCAL.NE.ONE ) THEN |
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373 GROW = ZERO |
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374 GO TO 80 |
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375 END IF |
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376 * |
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377 IF( NOUNIT ) THEN |
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378 * |
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379 * A is non-unit triangular. |
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380 * |
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381 * Compute GROW = 1/G(j) and XBND = 1/M(j). |
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382 * Initially, M(0) = max{x(i), i=1,...,n}. |
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383 * |
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384 GROW = ONE / MAX( XBND, SMLNUM ) |
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385 XBND = GROW |
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386 DO 60 J = JFIRST, JLAST, JINC |
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387 * |
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388 * Exit the loop if the growth factor is too small. |
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389 * |
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390 IF( GROW.LE.SMLNUM ) |
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391 $ GO TO 80 |
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392 * |
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393 * G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) ) |
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394 * |
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395 XJ = ONE + CNORM( J ) |
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396 GROW = MIN( GROW, XBND / XJ ) |
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397 * |
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398 * M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j)) |
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399 * |
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400 TJJ = ABS( AB( MAIND, J ) ) |
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401 IF( XJ.GT.TJJ ) |
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402 $ XBND = XBND*( TJJ / XJ ) |
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403 60 CONTINUE |
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404 GROW = MIN( GROW, XBND ) |
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405 ELSE |
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406 * |
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407 * A is unit triangular. |
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408 * |
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409 * Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}. |
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410 * |
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411 GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) ) |
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412 DO 70 J = JFIRST, JLAST, JINC |
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413 * |
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414 * Exit the loop if the growth factor is too small. |
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415 * |
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416 IF( GROW.LE.SMLNUM ) |
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417 $ GO TO 80 |
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418 * |
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419 * G(j) = ( 1 + CNORM(j) )*G(j-1) |
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420 * |
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421 XJ = ONE + CNORM( J ) |
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422 GROW = GROW / XJ |
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423 70 CONTINUE |
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424 END IF |
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425 80 CONTINUE |
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426 END IF |
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427 * |
|
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428 IF( ( GROW*TSCAL ).GT.SMLNUM ) THEN |
|
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429 * |
|
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430 * Use the Level 2 BLAS solve if the reciprocal of the bound on |
|
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431 * elements of X is not too small. |
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432 * |
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433 CALL DTBSV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, X, 1 ) |
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434 ELSE |
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435 * |
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436 * Use a Level 1 BLAS solve, scaling intermediate results. |
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437 * |
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438 IF( XMAX.GT.BIGNUM ) THEN |
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439 * |
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440 * Scale X so that its components are less than or equal to |
|
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441 * BIGNUM in absolute value. |
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442 * |
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443 SCALE = BIGNUM / XMAX |
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444 CALL DSCAL( N, SCALE, X, 1 ) |
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445 XMAX = BIGNUM |
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446 END IF |
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447 * |
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|
448 IF( NOTRAN ) THEN |
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449 * |
|
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450 * Solve A * x = b |
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451 * |
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452 DO 110 J = JFIRST, JLAST, JINC |
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453 * |
|
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454 * Compute x(j) = b(j) / A(j,j), scaling x if necessary. |
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455 * |
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456 XJ = ABS( X( J ) ) |
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457 IF( NOUNIT ) THEN |
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458 TJJS = AB( MAIND, J )*TSCAL |
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|
459 ELSE |
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460 TJJS = TSCAL |
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461 IF( TSCAL.EQ.ONE ) |
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462 $ GO TO 100 |
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463 END IF |
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464 TJJ = ABS( TJJS ) |
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465 IF( TJJ.GT.SMLNUM ) THEN |
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466 * |
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467 * abs(A(j,j)) > SMLNUM: |
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468 * |
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469 IF( TJJ.LT.ONE ) THEN |
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470 IF( XJ.GT.TJJ*BIGNUM ) THEN |
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471 * |
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472 * Scale x by 1/b(j). |
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473 * |
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474 REC = ONE / XJ |
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475 CALL DSCAL( N, REC, X, 1 ) |
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476 SCALE = SCALE*REC |
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477 XMAX = XMAX*REC |
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478 END IF |
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479 END IF |
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480 X( J ) = X( J ) / TJJS |
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481 XJ = ABS( X( J ) ) |
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482 ELSE IF( TJJ.GT.ZERO ) THEN |
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|
483 * |
|
|
484 * 0 < abs(A(j,j)) <= SMLNUM: |
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485 * |
|
|
486 IF( XJ.GT.TJJ*BIGNUM ) THEN |
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|
487 * |
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488 * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM |
|
|
489 * to avoid overflow when dividing by A(j,j). |
|
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490 * |
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491 REC = ( TJJ*BIGNUM ) / XJ |
|
|
492 IF( CNORM( J ).GT.ONE ) THEN |
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493 * |
|
|
494 * Scale by 1/CNORM(j) to avoid overflow when |
|
|
495 * multiplying x(j) times column j. |
|
|
496 * |
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497 REC = REC / CNORM( J ) |
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498 END IF |
|
|
499 CALL DSCAL( N, REC, X, 1 ) |
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500 SCALE = SCALE*REC |
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501 XMAX = XMAX*REC |
|
|
502 END IF |
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503 X( J ) = X( J ) / TJJS |
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|
504 XJ = ABS( X( J ) ) |
|
|
505 ELSE |
|
|
506 * |
|
|
507 * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and |
|
|
508 * scale = 0, and compute a solution to A*x = 0. |
|
|
509 * |
|
|
510 DO 90 I = 1, N |
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|
511 X( I ) = ZERO |
|
|
512 90 CONTINUE |
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513 X( J ) = ONE |
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|
514 XJ = ONE |
|
|
515 SCALE = ZERO |
|
|
516 XMAX = ZERO |
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|
517 END IF |
|
|
518 100 CONTINUE |
|
|
519 * |
|
|
520 * Scale x if necessary to avoid overflow when adding a |
|
|
521 * multiple of column j of A. |
|
|
522 * |
|
|
523 IF( XJ.GT.ONE ) THEN |
|
|
524 REC = ONE / XJ |
|
|
525 IF( CNORM( J ).GT.( BIGNUM-XMAX )*REC ) THEN |
|
|
526 * |
|
|
527 * Scale x by 1/(2*abs(x(j))). |
|
|
528 * |
|
|
529 REC = REC*HALF |
|
|
530 CALL DSCAL( N, REC, X, 1 ) |
|
|
531 SCALE = SCALE*REC |
|
|
532 END IF |
|
|
533 ELSE IF( XJ*CNORM( J ).GT.( BIGNUM-XMAX ) ) THEN |
|
|
534 * |
|
|
535 * Scale x by 1/2. |
|
|
536 * |
|
|
537 CALL DSCAL( N, HALF, X, 1 ) |
|
|
538 SCALE = SCALE*HALF |
|
|
539 END IF |
|
|
540 * |
|
|
541 IF( UPPER ) THEN |
|
|
542 IF( J.GT.1 ) THEN |
|
|
543 * |
|
|
544 * Compute the update |
|
|
545 * x(max(1,j-kd):j-1) := x(max(1,j-kd):j-1) - |
|
|
546 * x(j)* A(max(1,j-kd):j-1,j) |
|
|
547 * |
|
|
548 JLEN = MIN( KD, J-1 ) |
|
|
549 CALL DAXPY( JLEN, -X( J )*TSCAL, |
|
|
550 $ AB( KD+1-JLEN, J ), 1, X( J-JLEN ), 1 ) |
|
|
551 I = IDAMAX( J-1, X, 1 ) |
|
|
552 XMAX = ABS( X( I ) ) |
|
|
553 END IF |
|
|
554 ELSE IF( J.LT.N ) THEN |
|
|
555 * |
|
|
556 * Compute the update |
|
|
557 * x(j+1:min(j+kd,n)) := x(j+1:min(j+kd,n)) - |
|
|
558 * x(j) * A(j+1:min(j+kd,n),j) |
|
|
559 * |
|
|
560 JLEN = MIN( KD, N-J ) |
|
|
561 IF( JLEN.GT.0 ) |
|
|
562 $ CALL DAXPY( JLEN, -X( J )*TSCAL, AB( 2, J ), 1, |
|
|
563 $ X( J+1 ), 1 ) |
|
|
564 I = J + IDAMAX( N-J, X( J+1 ), 1 ) |
|
|
565 XMAX = ABS( X( I ) ) |
|
|
566 END IF |
|
|
567 110 CONTINUE |
|
|
568 * |
|
|
569 ELSE |
|
|
570 * |
|
|
571 * Solve A' * x = b |
|
|
572 * |
|
|
573 DO 160 J = JFIRST, JLAST, JINC |
|
|
574 * |
|
|
575 * Compute x(j) = b(j) - sum A(k,j)*x(k). |
|
|
576 * k<>j |
|
|
577 * |
|
|
578 XJ = ABS( X( J ) ) |
|
|
579 USCAL = TSCAL |
|
|
580 REC = ONE / MAX( XMAX, ONE ) |
|
|
581 IF( CNORM( J ).GT.( BIGNUM-XJ )*REC ) THEN |
|
|
582 * |
|
|
583 * If x(j) could overflow, scale x by 1/(2*XMAX). |
|
|
584 * |
|
|
585 REC = REC*HALF |
|
|
586 IF( NOUNIT ) THEN |
|
|
587 TJJS = AB( MAIND, J )*TSCAL |
|
|
588 ELSE |
|
|
589 TJJS = TSCAL |
|
|
590 END IF |
|
|
591 TJJ = ABS( TJJS ) |
|
|
592 IF( TJJ.GT.ONE ) THEN |
|
|
593 * |
|
|
594 * Divide by A(j,j) when scaling x if A(j,j) > 1. |
|
|
595 * |
|
|
596 REC = MIN( ONE, REC*TJJ ) |
|
|
597 USCAL = USCAL / TJJS |
|
|
598 END IF |
|
|
599 IF( REC.LT.ONE ) THEN |
|
|
600 CALL DSCAL( N, REC, X, 1 ) |
|
|
601 SCALE = SCALE*REC |
|
|
602 XMAX = XMAX*REC |
|
|
603 END IF |
|
|
604 END IF |
|
|
605 * |
|
|
606 SUMJ = ZERO |
|
|
607 IF( USCAL.EQ.ONE ) THEN |
|
|
608 * |
|
|
609 * If the scaling needed for A in the dot product is 1, |
|
|
610 * call DDOT to perform the dot product. |
|
|
611 * |
|
|
612 IF( UPPER ) THEN |
|
|
613 JLEN = MIN( KD, J-1 ) |
|
|
614 SUMJ = DDOT( JLEN, AB( KD+1-JLEN, J ), 1, |
|
|
615 $ X( J-JLEN ), 1 ) |
|
|
616 ELSE |
|
|
617 JLEN = MIN( KD, N-J ) |
|
|
618 IF( JLEN.GT.0 ) |
|
|
619 $ SUMJ = DDOT( JLEN, AB( 2, J ), 1, X( J+1 ), 1 ) |
|
|
620 END IF |
|
|
621 ELSE |
|
|
622 * |
|
|
623 * Otherwise, use in-line code for the dot product. |
|
|
624 * |
|
|
625 IF( UPPER ) THEN |
|
|
626 JLEN = MIN( KD, J-1 ) |
|
|
627 DO 120 I = 1, JLEN |
|
|
628 SUMJ = SUMJ + ( AB( KD+I-JLEN, J )*USCAL )* |
|
|
629 $ X( J-JLEN-1+I ) |
|
|
630 120 CONTINUE |
|
|
631 ELSE |
|
|
632 JLEN = MIN( KD, N-J ) |
|
|
633 DO 130 I = 1, JLEN |
|
|
634 SUMJ = SUMJ + ( AB( I+1, J )*USCAL )*X( J+I ) |
|
|
635 130 CONTINUE |
|
|
636 END IF |
|
|
637 END IF |
|
|
638 * |
|
|
639 IF( USCAL.EQ.TSCAL ) THEN |
|
|
640 * |
|
|
641 * Compute x(j) := ( x(j) - sumj ) / A(j,j) if 1/A(j,j) |
|
|
642 * was not used to scale the dotproduct. |
|
|
643 * |
|
|
644 X( J ) = X( J ) - SUMJ |
|
|
645 XJ = ABS( X( J ) ) |
|
|
646 IF( NOUNIT ) THEN |
|
|
647 * |
|
|
648 * Compute x(j) = x(j) / A(j,j), scaling if necessary. |
|
|
649 * |
|
|
650 TJJS = AB( MAIND, J )*TSCAL |
|
|
651 ELSE |
|
|
652 TJJS = TSCAL |
|
|
653 IF( TSCAL.EQ.ONE ) |
|
|
654 $ GO TO 150 |
|
|
655 END IF |
|
|
656 TJJ = ABS( TJJS ) |
|
|
657 IF( TJJ.GT.SMLNUM ) THEN |
|
|
658 * |
|
|
659 * abs(A(j,j)) > SMLNUM: |
|
|
660 * |
|
|
661 IF( TJJ.LT.ONE ) THEN |
|
|
662 IF( XJ.GT.TJJ*BIGNUM ) THEN |
|
|
663 * |
|
|
664 * Scale X by 1/abs(x(j)). |
|
|
665 * |
|
|
666 REC = ONE / XJ |
|
|
667 CALL DSCAL( N, REC, X, 1 ) |
|
|
668 SCALE = SCALE*REC |
|
|
669 XMAX = XMAX*REC |
|
|
670 END IF |
|
|
671 END IF |
|
|
672 X( J ) = X( J ) / TJJS |
|
|
673 ELSE IF( TJJ.GT.ZERO ) THEN |
|
|
674 * |
|
|
675 * 0 < abs(A(j,j)) <= SMLNUM: |
|
|
676 * |
|
|
677 IF( XJ.GT.TJJ*BIGNUM ) THEN |
|
|
678 * |
|
|
679 * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM. |
|
|
680 * |
|
|
681 REC = ( TJJ*BIGNUM ) / XJ |
|
|
682 CALL DSCAL( N, REC, X, 1 ) |
|
|
683 SCALE = SCALE*REC |
|
|
684 XMAX = XMAX*REC |
|
|
685 END IF |
|
|
686 X( J ) = X( J ) / TJJS |
|
|
687 ELSE |
|
|
688 * |
|
|
689 * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and |
|
|
690 * scale = 0, and compute a solution to A'*x = 0. |
|
|
691 * |
|
|
692 DO 140 I = 1, N |
|
|
693 X( I ) = ZERO |
|
|
694 140 CONTINUE |
|
|
695 X( J ) = ONE |
|
|
696 SCALE = ZERO |
|
|
697 XMAX = ZERO |
|
|
698 END IF |
|
|
699 150 CONTINUE |
|
|
700 ELSE |
|
|
701 * |
|
|
702 * Compute x(j) := x(j) / A(j,j) - sumj if the dot |
|
|
703 * product has already been divided by 1/A(j,j). |
|
|
704 * |
|
|
705 X( J ) = X( J ) / TJJS - SUMJ |
|
|
706 END IF |
|
|
707 XMAX = MAX( XMAX, ABS( X( J ) ) ) |
|
|
708 160 CONTINUE |
|
|
709 END IF |
|
|
710 SCALE = SCALE / TSCAL |
|
|
711 END IF |
|
|
712 * |
|
|
713 * Scale the column norms by 1/TSCAL for return. |
|
|
714 * |
|
|
715 IF( TSCAL.NE.ONE ) THEN |
|
|
716 CALL DSCAL( N, ONE / TSCAL, CNORM, 1 ) |
|
|
717 END IF |
|
|
718 * |
|
|
719 RETURN |
|
|
720 * |
|
|
721 * End of DLATBS |
|
|
722 * |
|
|
723 END |
