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1 SUBROUTINE DGEQRF( M, N, A, LDA, TAU, WORK, LWORK, INFO ) |
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2 * |
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3 * -- LAPACK routine (version 3.0) -- |
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4 * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., |
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5 * Courant Institute, Argonne National Lab, and Rice University |
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6 * June 30, 1999 |
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7 * |
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8 * .. Scalar Arguments .. |
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9 INTEGER INFO, LDA, LWORK, M, N |
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10 * .. |
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11 * .. Array Arguments .. |
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12 DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) |
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13 * .. |
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14 * |
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15 * Purpose |
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16 * ======= |
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17 * |
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18 * DGEQRF computes a QR factorization of a real M-by-N matrix A: |
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19 * A = Q * R. |
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20 * |
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21 * Arguments |
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22 * ========= |
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23 * |
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24 * M (input) INTEGER |
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25 * The number of rows of the matrix A. M >= 0. |
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26 * |
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27 * N (input) INTEGER |
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28 * The number of columns of the matrix A. N >= 0. |
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29 * |
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30 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) |
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31 * On entry, the M-by-N matrix A. |
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32 * On exit, the elements on and above the diagonal of the array |
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33 * contain the min(M,N)-by-N upper trapezoidal matrix R (R is |
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34 * upper triangular if m >= n); the elements below the diagonal, |
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35 * with the array TAU, represent the orthogonal matrix Q as a |
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36 * product of min(m,n) elementary reflectors (see Further |
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37 * Details). |
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38 * |
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39 * LDA (input) INTEGER |
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40 * The leading dimension of the array A. LDA >= max(1,M). |
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41 * |
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42 * TAU (output) DOUBLE PRECISION array, dimension (min(M,N)) |
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43 * The scalar factors of the elementary reflectors (see Further |
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44 * Details). |
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45 * |
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46 * WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) |
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47 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
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48 * |
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49 * LWORK (input) INTEGER |
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50 * The dimension of the array WORK. LWORK >= max(1,N). |
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51 * For optimum performance LWORK >= N*NB, where NB is |
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52 * the optimal blocksize. |
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53 * |
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54 * If LWORK = -1, then a workspace query is assumed; the routine |
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55 * only calculates the optimal size of the WORK array, returns |
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56 * this value as the first entry of the WORK array, and no error |
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57 * message related to LWORK is issued by XERBLA. |
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58 * |
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59 * INFO (output) INTEGER |
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60 * = 0: successful exit |
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61 * < 0: if INFO = -i, the i-th argument had an illegal value |
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62 * |
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63 * Further Details |
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64 * =============== |
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65 * |
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66 * The matrix Q is represented as a product of elementary reflectors |
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67 * |
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68 * Q = H(1) H(2) . . . H(k), where k = min(m,n). |
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69 * |
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70 * Each H(i) has the form |
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71 * |
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72 * H(i) = I - tau * v * v' |
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73 * |
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74 * where tau is a real scalar, and v is a real vector with |
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75 * v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), |
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76 * and tau in TAU(i). |
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77 * |
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78 * ===================================================================== |
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79 * |
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80 * .. Local Scalars .. |
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81 LOGICAL LQUERY |
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82 INTEGER I, IB, IINFO, IWS, K, LDWORK, LWKOPT, NB, |
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83 $ NBMIN, NX |
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84 * .. |
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85 * .. External Subroutines .. |
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86 EXTERNAL DGEQR2, DLARFB, DLARFT, XERBLA |
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87 * .. |
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88 * .. Intrinsic Functions .. |
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89 INTRINSIC MAX, MIN |
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90 * .. |
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91 * .. External Functions .. |
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92 INTEGER ILAENV |
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93 EXTERNAL ILAENV |
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94 * .. |
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95 * .. Executable Statements .. |
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96 * |
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97 * Test the input arguments |
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98 * |
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99 INFO = 0 |
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100 NB = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 ) |
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101 LWKOPT = N*NB |
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102 WORK( 1 ) = LWKOPT |
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103 LQUERY = ( LWORK.EQ.-1 ) |
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104 IF( M.LT.0 ) THEN |
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105 INFO = -1 |
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106 ELSE IF( N.LT.0 ) THEN |
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107 INFO = -2 |
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108 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN |
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109 INFO = -4 |
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110 ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN |
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111 INFO = -7 |
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112 END IF |
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113 IF( INFO.NE.0 ) THEN |
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114 CALL XERBLA( 'DGEQRF', -INFO ) |
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115 RETURN |
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116 ELSE IF( LQUERY ) THEN |
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117 RETURN |
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118 END IF |
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119 * |
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120 * Quick return if possible |
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121 * |
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122 K = MIN( M, N ) |
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123 IF( K.EQ.0 ) THEN |
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124 WORK( 1 ) = 1 |
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125 RETURN |
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126 END IF |
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127 * |
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128 NBMIN = 2 |
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129 NX = 0 |
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130 IWS = N |
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131 IF( NB.GT.1 .AND. NB.LT.K ) THEN |
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132 * |
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133 * Determine when to cross over from blocked to unblocked code. |
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134 * |
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135 NX = MAX( 0, ILAENV( 3, 'DGEQRF', ' ', M, N, -1, -1 ) ) |
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136 IF( NX.LT.K ) THEN |
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137 * |
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138 * Determine if workspace is large enough for blocked code. |
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139 * |
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140 LDWORK = N |
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141 IWS = LDWORK*NB |
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142 IF( LWORK.LT.IWS ) THEN |
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143 * |
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144 * Not enough workspace to use optimal NB: reduce NB and |
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145 * determine the minimum value of NB. |
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146 * |
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147 NB = LWORK / LDWORK |
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148 NBMIN = MAX( 2, ILAENV( 2, 'DGEQRF', ' ', M, N, -1, |
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149 $ -1 ) ) |
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150 END IF |
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151 END IF |
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152 END IF |
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153 * |
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154 IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN |
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155 * |
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156 * Use blocked code initially |
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157 * |
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158 DO 10 I = 1, K - NX, NB |
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159 IB = MIN( K-I+1, NB ) |
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160 * |
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161 * Compute the QR factorization of the current block |
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162 * A(i:m,i:i+ib-1) |
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163 * |
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164 CALL DGEQR2( M-I+1, IB, A( I, I ), LDA, TAU( I ), WORK, |
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165 $ IINFO ) |
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166 IF( I+IB.LE.N ) THEN |
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167 * |
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168 * Form the triangular factor of the block reflector |
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169 * H = H(i) H(i+1) . . . H(i+ib-1) |
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170 * |
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171 CALL DLARFT( 'Forward', 'Columnwise', M-I+1, IB, |
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172 $ A( I, I ), LDA, TAU( I ), WORK, LDWORK ) |
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173 * |
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174 * Apply H' to A(i:m,i+ib:n) from the left |
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175 * |
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176 CALL DLARFB( 'Left', 'Transpose', 'Forward', |
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177 $ 'Columnwise', M-I+1, N-I-IB+1, IB, |
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178 $ A( I, I ), LDA, WORK, LDWORK, A( I, I+IB ), |
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179 $ LDA, WORK( IB+1 ), LDWORK ) |
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180 END IF |
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181 10 CONTINUE |
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182 ELSE |
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183 I = 1 |
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184 END IF |
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185 * |
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186 * Use unblocked code to factor the last or only block. |
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187 * |
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188 IF( I.LE.K ) |
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189 $ CALL DGEQR2( M-I+1, N-I+1, A( I, I ), LDA, TAU( I ), WORK, |
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190 $ IINFO ) |
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191 * |
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192 WORK( 1 ) = IWS |
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193 RETURN |
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194 * |
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195 * End of DGEQRF |
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196 * |
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197 END |
