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1 SUBROUTINE DLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, |
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2 $ ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO ) |
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3 * |
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4 * -- LAPACK auxiliary routine (version 3.1) -- |
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5 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. |
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6 * November 2006 |
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7 * |
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8 * .. Scalar Arguments .. |
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9 INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N |
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10 LOGICAL WANTT, WANTZ |
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11 * .. |
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12 * .. Array Arguments .. |
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13 DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ), |
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14 $ Z( LDZ, * ) |
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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 * DLAQR0 computes the eigenvalues of a Hessenberg matrix H |
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21 * and, optionally, the matrices T and Z from the Schur decomposition |
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22 * H = Z T Z**T, where T is an upper quasi-triangular matrix (the |
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23 * Schur form), and Z is the orthogonal matrix of Schur vectors. |
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24 * |
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25 * Optionally Z may be postmultiplied into an input orthogonal |
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26 * matrix Q so that this routine can give the Schur factorization |
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27 * of a matrix A which has been reduced to the Hessenberg form H |
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28 * by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T. |
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29 * |
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30 * Arguments |
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31 * ========= |
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32 * |
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33 * WANTT (input) LOGICAL |
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34 * = .TRUE. : the full Schur form T is required; |
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35 * = .FALSE.: only eigenvalues are required. |
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36 * |
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37 * WANTZ (input) LOGICAL |
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38 * = .TRUE. : the matrix of Schur vectors Z is required; |
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39 * = .FALSE.: Schur vectors are not required. |
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40 * |
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41 * N (input) INTEGER |
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42 * The order of the matrix H. N .GE. 0. |
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43 * |
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44 * ILO (input) INTEGER |
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45 * IHI (input) INTEGER |
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46 * It is assumed that H is already upper triangular in rows |
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47 * and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1, |
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48 * H(ILO,ILO-1) is zero. ILO and IHI are normally set by a |
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49 * previous call to DGEBAL, and then passed to DGEHRD when the |
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50 * matrix output by DGEBAL is reduced to Hessenberg form. |
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51 * Otherwise, ILO and IHI should be set to 1 and N, |
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52 * respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N. |
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53 * If N = 0, then ILO = 1 and IHI = 0. |
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54 * |
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55 * H (input/output) DOUBLE PRECISION array, dimension (LDH,N) |
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56 * On entry, the upper Hessenberg matrix H. |
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57 * On exit, if INFO = 0 and WANTT is .TRUE., then H contains |
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58 * the upper quasi-triangular matrix T from the Schur |
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59 * decomposition (the Schur form); 2-by-2 diagonal blocks |
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60 * (corresponding to complex conjugate pairs of eigenvalues) |
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61 * are returned in standard form, with H(i,i) = H(i+1,i+1) |
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62 * and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is |
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63 * .FALSE., then the contents of H are unspecified on exit. |
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64 * (The output value of H when INFO.GT.0 is given under the |
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65 * description of INFO below.) |
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66 * |
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67 * This subroutine may explicitly set H(i,j) = 0 for i.GT.j and |
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68 * j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N. |
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69 * |
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70 * LDH (input) INTEGER |
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71 * The leading dimension of the array H. LDH .GE. max(1,N). |
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72 * |
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73 * WR (output) DOUBLE PRECISION array, dimension (IHI) |
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74 * WI (output) DOUBLE PRECISION array, dimension (IHI) |
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75 * The real and imaginary parts, respectively, of the computed |
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76 * eigenvalues of H(ILO:IHI,ILO:IHI) are stored WR(ILO:IHI) |
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77 * and WI(ILO:IHI). If two eigenvalues are computed as a |
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78 * complex conjugate pair, they are stored in consecutive |
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79 * elements of WR and WI, say the i-th and (i+1)th, with |
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80 * WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then |
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81 * the eigenvalues are stored in the same order as on the |
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82 * diagonal of the Schur form returned in H, with |
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83 * WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal |
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84 * block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and |
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85 * WI(i+1) = -WI(i). |
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86 * |
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87 * ILOZ (input) INTEGER |
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88 * IHIZ (input) INTEGER |
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89 * Specify the rows of Z to which transformations must be |
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90 * applied if WANTZ is .TRUE.. |
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91 * 1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N. |
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92 * |
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93 * Z (input/output) DOUBLE PRECISION array, dimension (LDZ,IHI) |
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94 * If WANTZ is .FALSE., then Z is not referenced. |
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95 * If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is |
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96 * replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the |
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97 * orthogonal Schur factor of H(ILO:IHI,ILO:IHI). |
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98 * (The output value of Z when INFO.GT.0 is given under |
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99 * the description of INFO below.) |
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100 * |
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101 * LDZ (input) INTEGER |
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102 * The leading dimension of the array Z. if WANTZ is .TRUE. |
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103 * then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1. |
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104 * |
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105 * WORK (workspace/output) DOUBLE PRECISION array, dimension LWORK |
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106 * On exit, if LWORK = -1, WORK(1) returns an estimate of |
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107 * the optimal value for LWORK. |
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108 * |
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109 * LWORK (input) INTEGER |
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110 * The dimension of the array WORK. LWORK .GE. max(1,N) |
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111 * is sufficient, but LWORK typically as large as 6*N may |
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112 * be required for optimal performance. A workspace query |
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113 * to determine the optimal workspace size is recommended. |
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114 * |
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115 * If LWORK = -1, then DLAQR0 does a workspace query. |
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116 * In this case, DLAQR0 checks the input parameters and |
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117 * estimates the optimal workspace size for the given |
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118 * values of N, ILO and IHI. The estimate is returned |
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119 * in WORK(1). No error message related to LWORK is |
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120 * issued by XERBLA. Neither H nor Z are accessed. |
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121 * |
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122 * |
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123 * INFO (output) INTEGER |
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124 * = 0: successful exit |
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125 * .GT. 0: if INFO = i, DLAQR0 failed to compute all of |
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126 * the eigenvalues. Elements 1:ilo-1 and i+1:n of WR |
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127 * and WI contain those eigenvalues which have been |
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128 * successfully computed. (Failures are rare.) |
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129 * |
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130 * If INFO .GT. 0 and WANT is .FALSE., then on exit, |
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131 * the remaining unconverged eigenvalues are the eigen- |
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132 * values of the upper Hessenberg matrix rows and |
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133 * columns ILO through INFO of the final, output |
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134 * value of H. |
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135 * |
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136 * If INFO .GT. 0 and WANTT is .TRUE., then on exit |
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137 * |
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138 * (*) (initial value of H)*U = U*(final value of H) |
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139 * |
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140 * where U is an orthogonal matrix. The final |
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141 * value of H is upper Hessenberg and quasi-triangular |
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142 * in rows and columns INFO+1 through IHI. |
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143 * |
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144 * If INFO .GT. 0 and WANTZ is .TRUE., then on exit |
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145 * |
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146 * (final value of Z(ILO:IHI,ILOZ:IHIZ) |
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147 * = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U |
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148 * |
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149 * where U is the orthogonal matrix in (*) (regard- |
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150 * less of the value of WANTT.) |
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151 * |
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152 * If INFO .GT. 0 and WANTZ is .FALSE., then Z is not |
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153 * accessed. |
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154 * |
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155 * |
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156 * ================================================================ |
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157 * Based on contributions by |
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158 * Karen Braman and Ralph Byers, Department of Mathematics, |
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159 * University of Kansas, USA |
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160 * |
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161 * ================================================================ |
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162 * |
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163 * References: |
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164 * K. Braman, R. Byers and R. Mathias, The Multi-Shift QR |
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165 * Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 |
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166 * Performance, SIAM Journal of Matrix Analysis, volume 23, pages |
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167 * 929--947, 2002. |
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168 * |
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169 * K. Braman, R. Byers and R. Mathias, The Multi-Shift QR |
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170 * Algorithm Part II: Aggressive Early Deflation, SIAM Journal |
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171 * of Matrix Analysis, volume 23, pages 948--973, 2002. |
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172 * |
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173 * ================================================================ |
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174 * .. Parameters .. |
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175 * |
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176 * ==== Matrices of order NTINY or smaller must be processed by |
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177 * . DLAHQR because of insufficient subdiagonal scratch space. |
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178 * . (This is a hard limit.) ==== |
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179 * |
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180 * ==== Exceptional deflation windows: try to cure rare |
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181 * . slow convergence by increasing the size of the |
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182 * . deflation window after KEXNW iterations. ===== |
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183 * |
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184 * ==== Exceptional shifts: try to cure rare slow convergence |
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185 * . with ad-hoc exceptional shifts every KEXSH iterations. |
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186 * . The constants WILK1 and WILK2 are used to form the |
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187 * . exceptional shifts. ==== |
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188 * |
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189 INTEGER NTINY |
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190 PARAMETER ( NTINY = 11 ) |
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191 INTEGER KEXNW, KEXSH |
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192 PARAMETER ( KEXNW = 5, KEXSH = 6 ) |
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193 DOUBLE PRECISION WILK1, WILK2 |
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194 PARAMETER ( WILK1 = 0.75d0, WILK2 = -0.4375d0 ) |
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195 DOUBLE PRECISION ZERO, ONE |
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196 PARAMETER ( ZERO = 0.0d0, ONE = 1.0d0 ) |
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197 * .. |
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198 * .. Local Scalars .. |
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199 DOUBLE PRECISION AA, BB, CC, CS, DD, SN, SS, SWAP |
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200 INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS, |
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201 $ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS, |
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202 $ LWKOPT, NDFL, NH, NHO, NIBBLE, NMIN, NS, NSMAX, |
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203 $ NSR, NVE, NW, NWMAX, NWR |
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204 LOGICAL NWINC, SORTED |
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205 CHARACTER JBCMPZ*2 |
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206 * .. |
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207 * .. External Functions .. |
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208 INTEGER ILAENV |
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209 EXTERNAL ILAENV |
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210 * .. |
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211 * .. Local Arrays .. |
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212 DOUBLE PRECISION ZDUM( 1, 1 ) |
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213 * .. |
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214 * .. External Subroutines .. |
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215 EXTERNAL DLACPY, DLAHQR, DLANV2, DLAQR3, DLAQR4, DLAQR5 |
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216 * .. |
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217 * .. Intrinsic Functions .. |
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218 INTRINSIC ABS, DBLE, INT, MAX, MIN, MOD |
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219 * .. |
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220 * .. Executable Statements .. |
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221 INFO = 0 |
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222 * |
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223 * ==== Quick return for N = 0: nothing to do. ==== |
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224 * |
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225 IF( N.EQ.0 ) THEN |
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226 WORK( 1 ) = ONE |
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227 RETURN |
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228 END IF |
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229 * |
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230 * ==== Set up job flags for ILAENV. ==== |
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231 * |
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232 IF( WANTT ) THEN |
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233 JBCMPZ( 1: 1 ) = 'S' |
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234 ELSE |
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235 JBCMPZ( 1: 1 ) = 'E' |
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236 END IF |
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237 IF( WANTZ ) THEN |
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238 JBCMPZ( 2: 2 ) = 'V' |
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239 ELSE |
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240 JBCMPZ( 2: 2 ) = 'N' |
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241 END IF |
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242 * |
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243 * ==== Tiny matrices must use DLAHQR. ==== |
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244 * |
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245 IF( N.LE.NTINY ) THEN |
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246 * |
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247 * ==== Estimate optimal workspace. ==== |
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248 * |
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249 LWKOPT = 1 |
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250 IF( LWORK.NE.-1 ) |
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251 $ CALL DLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, |
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252 $ ILOZ, IHIZ, Z, LDZ, INFO ) |
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253 ELSE |
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254 * |
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255 * ==== Use small bulge multi-shift QR with aggressive early |
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256 * . deflation on larger-than-tiny matrices. ==== |
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257 * |
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258 * ==== Hope for the best. ==== |
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259 * |
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260 INFO = 0 |
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261 * |
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262 * ==== NWR = recommended deflation window size. At this |
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263 * . point, N .GT. NTINY = 11, so there is enough |
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264 * . subdiagonal workspace for NWR.GE.2 as required. |
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265 * . (In fact, there is enough subdiagonal space for |
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266 * . NWR.GE.3.) ==== |
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267 * |
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268 NWR = ILAENV( 13, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) |
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269 NWR = MAX( 2, NWR ) |
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270 NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR ) |
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271 NW = NWR |
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272 * |
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273 * ==== NSR = recommended number of simultaneous shifts. |
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274 * . At this point N .GT. NTINY = 11, so there is at |
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275 * . enough subdiagonal workspace for NSR to be even |
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276 * . and greater than or equal to two as required. ==== |
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277 * |
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278 NSR = ILAENV( 15, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) |
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279 NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO ) |
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280 NSR = MAX( 2, NSR-MOD( NSR, 2 ) ) |
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281 * |
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282 * ==== Estimate optimal workspace ==== |
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283 * |
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284 * ==== Workspace query call to DLAQR3 ==== |
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285 * |
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286 CALL DLAQR3( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ, |
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287 $ IHIZ, Z, LDZ, LS, LD, WR, WI, H, LDH, N, H, LDH, |
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288 $ N, H, LDH, WORK, -1 ) |
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289 * |
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290 * ==== Optimal workspace = MAX(DLAQR5, DLAQR3) ==== |
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291 * |
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292 LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) ) |
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293 * |
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294 * ==== Quick return in case of workspace query. ==== |
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295 * |
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296 IF( LWORK.EQ.-1 ) THEN |
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297 WORK( 1 ) = DBLE( LWKOPT ) |
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298 RETURN |
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299 END IF |
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300 * |
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301 * ==== DLAHQR/DLAQR0 crossover point ==== |
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302 * |
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303 NMIN = ILAENV( 12, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) |
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304 NMIN = MAX( NTINY, NMIN ) |
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305 * |
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306 * ==== Nibble crossover point ==== |
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307 * |
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308 NIBBLE = ILAENV( 14, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) |
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309 NIBBLE = MAX( 0, NIBBLE ) |
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310 * |
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311 * ==== Accumulate reflections during ttswp? Use block |
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312 * . 2-by-2 structure during matrix-matrix multiply? ==== |
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313 * |
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314 KACC22 = ILAENV( 16, 'DLAQR0', JBCMPZ, N, ILO, IHI, LWORK ) |
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315 KACC22 = MAX( 0, KACC22 ) |
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316 KACC22 = MIN( 2, KACC22 ) |
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317 * |
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318 * ==== NWMAX = the largest possible deflation window for |
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319 * . which there is sufficient workspace. ==== |
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320 * |
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321 NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 ) |
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322 * |
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323 * ==== NSMAX = the Largest number of simultaneous shifts |
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324 * . for which there is sufficient workspace. ==== |
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325 * |
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326 NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 ) |
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327 NSMAX = NSMAX - MOD( NSMAX, 2 ) |
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328 * |
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329 * ==== NDFL: an iteration count restarted at deflation. ==== |
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330 * |
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331 NDFL = 1 |
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332 * |
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333 * ==== ITMAX = iteration limit ==== |
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334 * |
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335 ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) ) |
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336 * |
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337 * ==== Last row and column in the active block ==== |
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338 * |
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339 KBOT = IHI |
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340 * |
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341 * ==== Main Loop ==== |
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342 * |
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343 DO 80 IT = 1, ITMAX |
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344 * |
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345 * ==== Done when KBOT falls below ILO ==== |
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346 * |
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347 IF( KBOT.LT.ILO ) |
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348 $ GO TO 90 |
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349 * |
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350 * ==== Locate active block ==== |
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351 * |
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352 DO 10 K = KBOT, ILO + 1, -1 |
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353 IF( H( K, K-1 ).EQ.ZERO ) |
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354 $ GO TO 20 |
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355 10 CONTINUE |
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356 K = ILO |
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357 20 CONTINUE |
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358 KTOP = K |
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359 * |
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360 * ==== Select deflation window size ==== |
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361 * |
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362 NH = KBOT - KTOP + 1 |
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363 IF( NDFL.LT.KEXNW .OR. NH.LT.NW ) THEN |
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364 * |
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365 * ==== Typical deflation window. If possible and |
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366 * . advisable, nibble the entire active block. |
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367 * . If not, use size NWR or NWR+1 depending upon |
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368 * . which has the smaller corresponding subdiagonal |
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369 * . entry (a heuristic). ==== |
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370 * |
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371 NWINC = .TRUE. |
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372 IF( NH.LE.MIN( NMIN, NWMAX ) ) THEN |
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373 NW = NH |
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374 ELSE |
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375 NW = MIN( NWR, NH, NWMAX ) |
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376 IF( NW.LT.NWMAX ) THEN |
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377 IF( NW.GE.NH-1 ) THEN |
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378 NW = NH |
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379 ELSE |
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380 KWTOP = KBOT - NW + 1 |
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381 IF( ABS( H( KWTOP, KWTOP-1 ) ).GT. |
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382 $ ABS( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1 |
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383 END IF |
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384 END IF |
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385 END IF |
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386 ELSE |
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387 * |
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388 * ==== Exceptional deflation window. If there have |
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389 * . been no deflations in KEXNW or more iterations, |
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390 * . then vary the deflation window size. At first, |
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391 * . because, larger windows are, in general, more |
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392 * . powerful than smaller ones, rapidly increase the |
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393 * . window up to the maximum reasonable and possible. |
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394 * . Then maybe try a slightly smaller window. ==== |
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395 * |
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396 IF( NWINC .AND. NW.LT.MIN( NWMAX, NH ) ) THEN |
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397 NW = MIN( NWMAX, NH, 2*NW ) |
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398 ELSE |
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399 NWINC = .FALSE. |
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400 IF( NW.EQ.NH .AND. NH.GT.2 ) |
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401 $ NW = NH - 1 |
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402 END IF |
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403 END IF |
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404 * |
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405 * ==== Aggressive early deflation: |
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406 * . split workspace under the subdiagonal into |
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407 * . - an nw-by-nw work array V in the lower |
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408 * . left-hand-corner, |
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409 * . - an NW-by-at-least-NW-but-more-is-better |
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410 * . (NW-by-NHO) horizontal work array along |
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411 * . the bottom edge, |
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412 * . - an at-least-NW-but-more-is-better (NHV-by-NW) |
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413 * . vertical work array along the left-hand-edge. |
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414 * . ==== |
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415 * |
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416 KV = N - NW + 1 |
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417 KT = NW + 1 |
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418 NHO = ( N-NW-1 ) - KT + 1 |
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419 KWV = NW + 2 |
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420 NVE = ( N-NW ) - KWV + 1 |
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421 * |
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422 * ==== Aggressive early deflation ==== |
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423 * |
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424 CALL DLAQR3( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, |
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425 $ IHIZ, Z, LDZ, LS, LD, WR, WI, H( KV, 1 ), LDH, |
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426 $ NHO, H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH, |
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427 $ WORK, LWORK ) |
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428 * |
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429 * ==== Adjust KBOT accounting for new deflations. ==== |
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430 * |
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431 KBOT = KBOT - LD |
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432 * |
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433 * ==== KS points to the shifts. ==== |
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434 * |
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435 KS = KBOT - LS + 1 |
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436 * |
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437 * ==== Skip an expensive QR sweep if there is a (partly |
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438 * . heuristic) reason to expect that many eigenvalues |
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439 * . will deflate without it. Here, the QR sweep is |
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440 * . skipped if many eigenvalues have just been deflated |
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441 * . or if the remaining active block is small. |
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442 * |
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443 IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT- |
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444 $ KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN |
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445 * |
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446 * ==== NS = nominal number of simultaneous shifts. |
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447 * . This may be lowered (slightly) if DLAQR3 |
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448 * . did not provide that many shifts. ==== |
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449 * |
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450 NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) ) |
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451 NS = NS - MOD( NS, 2 ) |
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452 * |
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453 * ==== If there have been no deflations |
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454 * . in a multiple of KEXSH iterations, |
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455 * . then try exceptional shifts. |
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456 * . Otherwise use shifts provided by |
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457 * . DLAQR3 above or from the eigenvalues |
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458 * . of a trailing principal submatrix. ==== |
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459 * |
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460 IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN |
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461 KS = KBOT - NS + 1 |
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462 DO 30 I = KBOT, MAX( KS+1, KTOP+2 ), -2 |
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463 SS = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) ) |
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464 AA = WILK1*SS + H( I, I ) |
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465 BB = SS |
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466 CC = WILK2*SS |
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467 DD = AA |
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468 CALL DLANV2( AA, BB, CC, DD, WR( I-1 ), WI( I-1 ), |
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469 $ WR( I ), WI( I ), CS, SN ) |
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470 30 CONTINUE |
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471 IF( KS.EQ.KTOP ) THEN |
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472 WR( KS+1 ) = H( KS+1, KS+1 ) |
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473 WI( KS+1 ) = ZERO |
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474 WR( KS ) = WR( KS+1 ) |
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475 WI( KS ) = WI( KS+1 ) |
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476 END IF |
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477 ELSE |
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478 * |
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479 * ==== Got NS/2 or fewer shifts? Use DLAQR4 or |
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480 * . DLAHQR on a trailing principal submatrix to |
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481 * . get more. (Since NS.LE.NSMAX.LE.(N+6)/9, |
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482 * . there is enough space below the subdiagonal |
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483 * . to fit an NS-by-NS scratch array.) ==== |
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484 * |
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485 IF( KBOT-KS+1.LE.NS / 2 ) THEN |
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486 KS = KBOT - NS + 1 |
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487 KT = N - NS + 1 |
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488 CALL DLACPY( 'A', NS, NS, H( KS, KS ), LDH, |
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489 $ H( KT, 1 ), LDH ) |
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490 IF( NS.GT.NMIN ) THEN |
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491 CALL DLAQR4( .false., .false., NS, 1, NS, |
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492 $ H( KT, 1 ), LDH, WR( KS ), |
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493 $ WI( KS ), 1, 1, ZDUM, 1, WORK, |
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494 $ LWORK, INF ) |
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495 ELSE |
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496 CALL DLAHQR( .false., .false., NS, 1, NS, |
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497 $ H( KT, 1 ), LDH, WR( KS ), |
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498 $ WI( KS ), 1, 1, ZDUM, 1, INF ) |
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499 END IF |
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500 KS = KS + INF |
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501 * |
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502 * ==== In case of a rare QR failure use |
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503 * . eigenvalues of the trailing 2-by-2 |
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504 * . principal submatrix. ==== |
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505 * |
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506 IF( KS.GE.KBOT ) THEN |
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507 AA = H( KBOT-1, KBOT-1 ) |
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508 CC = H( KBOT, KBOT-1 ) |
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509 BB = H( KBOT-1, KBOT ) |
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510 DD = H( KBOT, KBOT ) |
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511 CALL DLANV2( AA, BB, CC, DD, WR( KBOT-1 ), |
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512 $ WI( KBOT-1 ), WR( KBOT ), |
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513 $ WI( KBOT ), CS, SN ) |
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514 KS = KBOT - 1 |
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515 END IF |
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516 END IF |
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517 * |
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518 IF( KBOT-KS+1.GT.NS ) THEN |
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519 * |
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520 * ==== Sort the shifts (Helps a little) |
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|
521 * . Bubble sort keeps complex conjugate |
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|
522 * . pairs together. ==== |
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523 * |
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524 SORTED = .false. |
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525 DO 50 K = KBOT, KS + 1, -1 |
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526 IF( SORTED ) |
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527 $ GO TO 60 |
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528 SORTED = .true. |
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529 DO 40 I = KS, K - 1 |
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530 IF( ABS( WR( I ) )+ABS( WI( I ) ).LT. |
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531 $ ABS( WR( I+1 ) )+ABS( WI( I+1 ) ) ) THEN |
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532 SORTED = .false. |
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|
533 * |
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|
534 SWAP = WR( I ) |
|
|
535 WR( I ) = WR( I+1 ) |
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|
536 WR( I+1 ) = SWAP |
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|
537 * |
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|
538 SWAP = WI( I ) |
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|
539 WI( I ) = WI( I+1 ) |
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|
540 WI( I+1 ) = SWAP |
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541 END IF |
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|
542 40 CONTINUE |
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543 50 CONTINUE |
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544 60 CONTINUE |
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545 END IF |
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546 * |
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|
547 * ==== Shuffle shifts into pairs of real shifts |
|
|
548 * . and pairs of complex conjugate shifts |
|
|
549 * . assuming complex conjugate shifts are |
|
|
550 * . already adjacent to one another. (Yes, |
|
|
551 * . they are.) ==== |
|
|
552 * |
|
|
553 DO 70 I = KBOT, KS + 2, -2 |
|
|
554 IF( WI( I ).NE.-WI( I-1 ) ) THEN |
|
|
555 * |
|
|
556 SWAP = WR( I ) |
|
|
557 WR( I ) = WR( I-1 ) |
|
|
558 WR( I-1 ) = WR( I-2 ) |
|
|
559 WR( I-2 ) = SWAP |
|
|
560 * |
|
|
561 SWAP = WI( I ) |
|
|
562 WI( I ) = WI( I-1 ) |
|
|
563 WI( I-1 ) = WI( I-2 ) |
|
|
564 WI( I-2 ) = SWAP |
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|
565 END IF |
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|
566 70 CONTINUE |
|
|
567 END IF |
|
|
568 * |
|
|
569 * ==== If there are only two shifts and both are |
|
|
570 * . real, then use only one. ==== |
|
|
571 * |
|
|
572 IF( KBOT-KS+1.EQ.2 ) THEN |
|
|
573 IF( WI( KBOT ).EQ.ZERO ) THEN |
|
|
574 IF( ABS( WR( KBOT )-H( KBOT, KBOT ) ).LT. |
|
|
575 $ ABS( WR( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN |
|
|
576 WR( KBOT-1 ) = WR( KBOT ) |
|
|
577 ELSE |
|
|
578 WR( KBOT ) = WR( KBOT-1 ) |
|
|
579 END IF |
|
|
580 END IF |
|
|
581 END IF |
|
|
582 * |
|
|
583 * ==== Use up to NS of the the smallest magnatiude |
|
|
584 * . shifts. If there aren't NS shifts available, |
|
|
585 * . then use them all, possibly dropping one to |
|
|
586 * . make the number of shifts even. ==== |
|
|
587 * |
|
|
588 NS = MIN( NS, KBOT-KS+1 ) |
|
|
589 NS = NS - MOD( NS, 2 ) |
|
|
590 KS = KBOT - NS + 1 |
|
|
591 * |
|
|
592 * ==== Small-bulge multi-shift QR sweep: |
|
|
593 * . split workspace under the subdiagonal into |
|
|
594 * . - a KDU-by-KDU work array U in the lower |
|
|
595 * . left-hand-corner, |
|
|
596 * . - a KDU-by-at-least-KDU-but-more-is-better |
|
|
597 * . (KDU-by-NHo) horizontal work array WH along |
|
|
598 * . the bottom edge, |
|
|
599 * . - and an at-least-KDU-but-more-is-better-by-KDU |
|
|
600 * . (NVE-by-KDU) vertical work WV arrow along |
|
|
601 * . the left-hand-edge. ==== |
|
|
602 * |
|
|
603 KDU = 3*NS - 3 |
|
|
604 KU = N - KDU + 1 |
|
|
605 KWH = KDU + 1 |
|
|
606 NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1 |
|
|
607 KWV = KDU + 4 |
|
|
608 NVE = N - KDU - KWV + 1 |
|
|
609 * |
|
|
610 * ==== Small-bulge multi-shift QR sweep ==== |
|
|
611 * |
|
|
612 CALL DLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS, |
|
|
613 $ WR( KS ), WI( KS ), H, LDH, ILOZ, IHIZ, Z, |
|
|
614 $ LDZ, WORK, 3, H( KU, 1 ), LDH, NVE, |
|
|
615 $ H( KWV, 1 ), LDH, NHO, H( KU, KWH ), LDH ) |
|
|
616 END IF |
|
|
617 * |
|
|
618 * ==== Note progress (or the lack of it). ==== |
|
|
619 * |
|
|
620 IF( LD.GT.0 ) THEN |
|
|
621 NDFL = 1 |
|
|
622 ELSE |
|
|
623 NDFL = NDFL + 1 |
|
|
624 END IF |
|
|
625 * |
|
|
626 * ==== End of main loop ==== |
|
|
627 80 CONTINUE |
|
|
628 * |
|
|
629 * ==== Iteration limit exceeded. Set INFO to show where |
|
|
630 * . the problem occurred and exit. ==== |
|
|
631 * |
|
|
632 INFO = KBOT |
|
|
633 90 CONTINUE |
|
|
634 END IF |
|
|
635 * |
|
|
636 * ==== Return the optimal value of LWORK. ==== |
|
|
637 * |
|
|
638 WORK( 1 ) = DBLE( LWKOPT ) |
|
|
639 * |
|
|
640 * ==== End of DLAQR0 ==== |
|
|
641 * |
|
|
642 END |
