1 | *> \brief \b SLATRZ |
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2 | * |
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3 | * =========== DOCUMENTATION =========== |
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4 | * |
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5 | * Online html documentation available at |
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6 | * http://www.netlib.org/lapack/explore-html/ |
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7 | * |
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8 | *> \htmlonly |
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9 | *> Download SLATRZ + dependencies |
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10 | *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slatrz.f"> |
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11 | *> [TGZ]</a> |
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12 | *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slatrz.f"> |
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13 | *> [ZIP]</a> |
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14 | *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slatrz.f"> |
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15 | *> [TXT]</a> |
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16 | *> \endhtmlonly |
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17 | * |
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18 | * Definition: |
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19 | * =========== |
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20 | * |
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21 | * SUBROUTINE SLATRZ( M, N, L, A, LDA, TAU, WORK ) |
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22 | * |
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23 | * .. Scalar Arguments .. |
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24 | * INTEGER L, LDA, M, N |
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25 | * .. |
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26 | * .. Array Arguments .. |
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27 | * REAL A( LDA, * ), TAU( * ), WORK( * ) |
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28 | * .. |
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29 | * |
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30 | * |
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31 | *> \par Purpose: |
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32 | * ============= |
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33 | *> |
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34 | *> \verbatim |
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35 | *> |
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36 | *> SLATRZ factors the M-by-(M+L) real upper trapezoidal matrix |
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37 | *> [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R 0 ) * Z, by means |
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38 | *> of orthogonal transformations. Z is an (M+L)-by-(M+L) orthogonal |
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39 | *> matrix and, R and A1 are M-by-M upper triangular matrices. |
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40 | *> \endverbatim |
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41 | * |
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42 | * Arguments: |
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43 | * ========== |
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44 | * |
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45 | *> \param[in] M |
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46 | *> \verbatim |
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47 | *> M is INTEGER |
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48 | *> The number of rows of the matrix A. M >= 0. |
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49 | *> \endverbatim |
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50 | *> |
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51 | *> \param[in] N |
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52 | *> \verbatim |
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53 | *> N is INTEGER |
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54 | *> The number of columns of the matrix A. N >= 0. |
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55 | *> \endverbatim |
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56 | *> |
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57 | *> \param[in] L |
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58 | *> \verbatim |
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59 | *> L is INTEGER |
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60 | *> The number of columns of the matrix A containing the |
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61 | *> meaningful part of the Householder vectors. N-M >= L >= 0. |
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62 | *> \endverbatim |
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63 | *> |
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64 | *> \param[in,out] A |
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65 | *> \verbatim |
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66 | *> A is REAL array, dimension (LDA,N) |
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67 | *> On entry, the leading M-by-N upper trapezoidal part of the |
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68 | *> array A must contain the matrix to be factorized. |
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69 | *> On exit, the leading M-by-M upper triangular part of A |
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70 | *> contains the upper triangular matrix R, and elements N-L+1 to |
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71 | *> N of the first M rows of A, with the array TAU, represent the |
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72 | *> orthogonal matrix Z as a product of M elementary reflectors. |
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73 | *> \endverbatim |
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74 | *> |
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75 | *> \param[in] LDA |
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76 | *> \verbatim |
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77 | *> LDA is INTEGER |
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78 | *> The leading dimension of the array A. LDA >= max(1,M). |
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79 | *> \endverbatim |
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80 | *> |
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81 | *> \param[out] TAU |
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82 | *> \verbatim |
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83 | *> TAU is REAL array, dimension (M) |
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84 | *> The scalar factors of the elementary reflectors. |
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85 | *> \endverbatim |
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86 | *> |
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87 | *> \param[out] WORK |
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88 | *> \verbatim |
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89 | *> WORK is REAL array, dimension (M) |
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90 | *> \endverbatim |
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91 | * |
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92 | * Authors: |
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93 | * ======== |
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94 | * |
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95 | *> \author Univ. of Tennessee |
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96 | *> \author Univ. of California Berkeley |
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97 | *> \author Univ. of Colorado Denver |
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98 | *> \author NAG Ltd. |
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99 | * |
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100 | *> \date November 2011 |
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101 | * |
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102 | *> \ingroup realOTHERcomputational |
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103 | * |
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104 | *> \par Contributors: |
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105 | * ================== |
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106 | *> |
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107 | *> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA |
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108 | * |
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109 | *> \par Further Details: |
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110 | * ===================== |
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111 | *> |
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112 | *> \verbatim |
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113 | *> |
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114 | *> The factorization is obtained by Householder's method. The kth |
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115 | *> transformation matrix, Z( k ), which is used to introduce zeros into |
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116 | *> the ( m - k + 1 )th row of A, is given in the form |
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117 | *> |
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118 | *> Z( k ) = ( I 0 ), |
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119 | *> ( 0 T( k ) ) |
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120 | *> |
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121 | *> where |
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122 | *> |
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123 | *> T( k ) = I - tau*u( k )*u( k )**T, u( k ) = ( 1 ), |
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124 | *> ( 0 ) |
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125 | *> ( z( k ) ) |
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126 | *> |
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127 | *> tau is a scalar and z( k ) is an l element vector. tau and z( k ) |
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128 | *> are chosen to annihilate the elements of the kth row of A2. |
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129 | *> |
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130 | *> The scalar tau is returned in the kth element of TAU and the vector |
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131 | *> u( k ) in the kth row of A2, such that the elements of z( k ) are |
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132 | *> in a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in |
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133 | *> the upper triangular part of A1. |
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134 | *> |
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135 | *> Z is given by |
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136 | *> |
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137 | *> Z = Z( 1 ) * Z( 2 ) * ... * Z( m ). |
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138 | *> \endverbatim |
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139 | *> |
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140 | * ===================================================================== |
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141 | SUBROUTINE SLATRZ( M, N, L, A, LDA, TAU, WORK ) |
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142 | * |
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143 | * -- LAPACK computational routine (version 3.4.0) -- |
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144 | * -- LAPACK is a software package provided by Univ. of Tennessee, -- |
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145 | * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
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146 | * November 2011 |
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147 | * |
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148 | * .. Scalar Arguments .. |
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149 | INTEGER L, LDA, M, N |
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150 | * .. |
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151 | * .. Array Arguments .. |
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152 | REAL A( LDA, * ), TAU( * ), WORK( * ) |
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153 | * .. |
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154 | * |
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155 | * ===================================================================== |
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156 | * |
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157 | * .. Parameters .. |
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158 | REAL ZERO |
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159 | PARAMETER ( ZERO = 0.0E+0 ) |
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160 | * .. |
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161 | * .. Local Scalars .. |
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162 | INTEGER I |
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163 | * .. |
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164 | * .. External Subroutines .. |
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165 | EXTERNAL SLARFG, SLARZ |
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166 | * .. |
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167 | * .. Executable Statements .. |
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168 | * |
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169 | * Test the input arguments |
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170 | * |
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171 | * Quick return if possible |
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172 | * |
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173 | IF( M.EQ.0 ) THEN |
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174 | RETURN |
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175 | ELSE IF( M.EQ.N ) THEN |
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176 | DO 10 I = 1, N |
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177 | TAU( I ) = ZERO |
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178 | 10 CONTINUE |
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179 | RETURN |
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180 | END IF |
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181 | * |
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182 | DO 20 I = M, 1, -1 |
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183 | * |
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184 | * Generate elementary reflector H(i) to annihilate |
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185 | * [ A(i,i) A(i,n-l+1:n) ] |
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186 | * |
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187 | CALL SLARFG( L+1, A( I, I ), A( I, N-L+1 ), LDA, TAU( I ) ) |
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188 | * |
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189 | * Apply H(i) to A(1:i-1,i:n) from the right |
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190 | * |
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191 | CALL SLARZ( 'Right', I-1, N-I+1, L, A( I, N-L+1 ), LDA, |
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192 | $ TAU( I ), A( 1, I ), LDA, WORK ) |
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193 | * |
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194 | 20 CONTINUE |
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195 | * |
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196 | RETURN |
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197 | * |
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198 | * End of SLATRZ |
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199 | * |
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200 | END |
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