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1 SUBROUTINE DSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO ) |
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2 * |
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3 * -- LAPACK routine (version 3.1) -- |
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4 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. |
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5 * November 2006 |
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6 * |
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7 * .. Scalar Arguments .. |
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8 CHARACTER UPLO |
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9 INTEGER INFO, LDA, LWORK, N |
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10 * .. |
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11 * .. Array Arguments .. |
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12 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ), |
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13 $ WORK( * ) |
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14 * .. |
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15 * |
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16 * Purpose |
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17 * ======= |
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18 * |
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19 * DSYTRD reduces a real symmetric matrix A to real symmetric |
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20 * tridiagonal form T by an orthogonal similarity transformation: |
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21 * Q**T * A * Q = T. |
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22 * |
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23 * Arguments |
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24 * ========= |
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25 * |
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26 * UPLO (input) CHARACTER*1 |
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27 * = 'U': Upper triangle of A is stored; |
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28 * = 'L': Lower triangle of A is stored. |
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29 * |
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30 * N (input) INTEGER |
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31 * The order of the matrix A. N >= 0. |
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32 * |
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33 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) |
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34 * On entry, the symmetric matrix A. If UPLO = 'U', the leading |
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35 * N-by-N upper triangular part of A contains the upper |
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36 * triangular part of the matrix A, and the strictly lower |
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37 * triangular part of A is not referenced. If UPLO = 'L', the |
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38 * leading N-by-N lower triangular part of A contains the lower |
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39 * triangular part of the matrix A, and the strictly upper |
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40 * triangular part of A is not referenced. |
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41 * On exit, if UPLO = 'U', the diagonal and first superdiagonal |
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42 * of A are overwritten by the corresponding elements of the |
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43 * tridiagonal matrix T, and the elements above the first |
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44 * superdiagonal, with the array TAU, represent the orthogonal |
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45 * matrix Q as a product of elementary reflectors; if UPLO |
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46 * = 'L', the diagonal and first subdiagonal of A are over- |
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47 * written by the corresponding elements of the tridiagonal |
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48 * matrix T, and the elements below the first subdiagonal, with |
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49 * the array TAU, represent the orthogonal matrix Q as a product |
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50 * of elementary reflectors. See Further Details. |
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51 * |
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52 * LDA (input) INTEGER |
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53 * The leading dimension of the array A. LDA >= max(1,N). |
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54 * |
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55 * D (output) DOUBLE PRECISION array, dimension (N) |
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56 * The diagonal elements of the tridiagonal matrix T: |
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57 * D(i) = A(i,i). |
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58 * |
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59 * E (output) DOUBLE PRECISION array, dimension (N-1) |
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60 * The off-diagonal elements of the tridiagonal matrix T: |
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61 * E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. |
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62 * |
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63 * TAU (output) DOUBLE PRECISION array, dimension (N-1) |
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64 * The scalar factors of the elementary reflectors (see Further |
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65 * Details). |
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66 * |
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67 * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) |
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68 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
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69 * |
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70 * LWORK (input) INTEGER |
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71 * The dimension of the array WORK. LWORK >= 1. |
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72 * For optimum performance LWORK >= N*NB, where NB is the |
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73 * optimal blocksize. |
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74 * |
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75 * If LWORK = -1, then a workspace query is assumed; the routine |
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76 * only calculates the optimal size of the WORK array, returns |
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77 * this value as the first entry of the WORK array, and no error |
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78 * message related to LWORK is issued by XERBLA. |
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79 * |
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80 * INFO (output) INTEGER |
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81 * = 0: successful exit |
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82 * < 0: if INFO = -i, the i-th argument had an illegal value |
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83 * |
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84 * Further Details |
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85 * =============== |
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86 * |
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87 * If UPLO = 'U', the matrix Q is represented as a product of elementary |
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88 * reflectors |
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89 * |
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90 * Q = H(n-1) . . . H(2) H(1). |
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91 * |
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92 * Each H(i) has the form |
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93 * |
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94 * H(i) = I - tau * v * v' |
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95 * |
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96 * where tau is a real scalar, and v is a real vector with |
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97 * v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in |
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98 * A(1:i-1,i+1), and tau in TAU(i). |
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99 * |
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100 * If UPLO = 'L', the matrix Q is represented as a product of elementary |
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101 * reflectors |
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102 * |
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103 * Q = H(1) H(2) . . . H(n-1). |
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104 * |
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105 * Each H(i) has the form |
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106 * |
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107 * H(i) = I - tau * v * v' |
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108 * |
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109 * where tau is a real scalar, and v is a real vector with |
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110 * v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), |
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111 * and tau in TAU(i). |
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112 * |
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113 * The contents of A on exit are illustrated by the following examples |
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114 * with n = 5: |
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115 * |
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116 * if UPLO = 'U': if UPLO = 'L': |
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117 * |
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118 * ( d e v2 v3 v4 ) ( d ) |
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119 * ( d e v3 v4 ) ( e d ) |
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120 * ( d e v4 ) ( v1 e d ) |
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121 * ( d e ) ( v1 v2 e d ) |
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122 * ( d ) ( v1 v2 v3 e d ) |
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123 * |
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124 * where d and e denote diagonal and off-diagonal elements of T, and vi |
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125 * denotes an element of the vector defining H(i). |
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126 * |
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127 * ===================================================================== |
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128 * |
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129 * .. Parameters .. |
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130 DOUBLE PRECISION ONE |
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131 PARAMETER ( ONE = 1.0D+0 ) |
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132 * .. |
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133 * .. Local Scalars .. |
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134 LOGICAL LQUERY, UPPER |
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135 INTEGER I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB, |
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136 $ NBMIN, NX |
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137 * .. |
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138 * .. External Subroutines .. |
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139 EXTERNAL DLATRD, DSYR2K, DSYTD2, XERBLA |
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140 * .. |
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141 * .. Intrinsic Functions .. |
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142 INTRINSIC MAX |
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143 * .. |
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144 * .. External Functions .. |
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145 LOGICAL LSAME |
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146 INTEGER ILAENV |
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147 EXTERNAL LSAME, ILAENV |
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148 * .. |
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149 * .. Executable Statements .. |
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150 * |
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151 * Test the input parameters |
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152 * |
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153 INFO = 0 |
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154 UPPER = LSAME( UPLO, 'U' ) |
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155 LQUERY = ( LWORK.EQ.-1 ) |
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156 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN |
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157 INFO = -1 |
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158 ELSE IF( N.LT.0 ) THEN |
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159 INFO = -2 |
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160 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN |
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161 INFO = -4 |
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162 ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN |
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163 INFO = -9 |
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164 END IF |
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165 * |
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166 IF( INFO.EQ.0 ) THEN |
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167 * |
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168 * Determine the block size. |
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169 * |
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170 NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 ) |
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171 LWKOPT = N*NB |
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172 WORK( 1 ) = LWKOPT |
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173 END IF |
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174 * |
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175 IF( INFO.NE.0 ) THEN |
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176 CALL XERBLA( 'DSYTRD', -INFO ) |
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177 RETURN |
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178 ELSE IF( LQUERY ) THEN |
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179 RETURN |
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180 END IF |
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181 * |
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182 * Quick return if possible |
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183 * |
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184 IF( N.EQ.0 ) THEN |
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185 WORK( 1 ) = 1 |
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186 RETURN |
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187 END IF |
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188 * |
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189 NX = N |
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190 IWS = 1 |
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191 IF( NB.GT.1 .AND. NB.LT.N ) THEN |
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192 * |
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193 * Determine when to cross over from blocked to unblocked code |
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194 * (last block is always handled by unblocked code). |
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195 * |
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196 NX = MAX( NB, ILAENV( 3, 'DSYTRD', UPLO, N, -1, -1, -1 ) ) |
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197 IF( NX.LT.N ) THEN |
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198 * |
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199 * Determine if workspace is large enough for blocked code. |
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200 * |
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201 LDWORK = N |
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202 IWS = LDWORK*NB |
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203 IF( LWORK.LT.IWS ) THEN |
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204 * |
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205 * Not enough workspace to use optimal NB: determine the |
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206 * minimum value of NB, and reduce NB or force use of |
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207 * unblocked code by setting NX = N. |
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208 * |
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209 NB = MAX( LWORK / LDWORK, 1 ) |
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210 NBMIN = ILAENV( 2, 'DSYTRD', UPLO, N, -1, -1, -1 ) |
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211 IF( NB.LT.NBMIN ) |
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212 $ NX = N |
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213 END IF |
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214 ELSE |
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215 NX = N |
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216 END IF |
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217 ELSE |
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218 NB = 1 |
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219 END IF |
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220 * |
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221 IF( UPPER ) THEN |
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222 * |
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223 * Reduce the upper triangle of A. |
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224 * Columns 1:kk are handled by the unblocked method. |
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225 * |
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226 KK = N - ( ( N-NX+NB-1 ) / NB )*NB |
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227 DO 20 I = N - NB + 1, KK + 1, -NB |
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228 * |
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229 * Reduce columns i:i+nb-1 to tridiagonal form and form the |
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230 * matrix W which is needed to update the unreduced part of |
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231 * the matrix |
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232 * |
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233 CALL DLATRD( UPLO, I+NB-1, NB, A, LDA, E, TAU, WORK, |
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234 $ LDWORK ) |
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235 * |
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236 * Update the unreduced submatrix A(1:i-1,1:i-1), using an |
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237 * update of the form: A := A - V*W' - W*V' |
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238 * |
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239 CALL DSYR2K( UPLO, 'No transpose', I-1, NB, -ONE, A( 1, I ), |
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240 $ LDA, WORK, LDWORK, ONE, A, LDA ) |
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241 * |
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242 * Copy superdiagonal elements back into A, and diagonal |
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243 * elements into D |
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244 * |
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245 DO 10 J = I, I + NB - 1 |
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246 A( J-1, J ) = E( J-1 ) |
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247 D( J ) = A( J, J ) |
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248 10 CONTINUE |
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249 20 CONTINUE |
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250 * |
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251 * Use unblocked code to reduce the last or only block |
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252 * |
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253 CALL DSYTD2( UPLO, KK, A, LDA, D, E, TAU, IINFO ) |
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254 ELSE |
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255 * |
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256 * Reduce the lower triangle of A |
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257 * |
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258 DO 40 I = 1, N - NX, NB |
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259 * |
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260 * Reduce columns i:i+nb-1 to tridiagonal form and form the |
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261 * matrix W which is needed to update the unreduced part of |
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262 * the matrix |
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263 * |
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264 CALL DLATRD( UPLO, N-I+1, NB, A( I, I ), LDA, E( I ), |
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265 $ TAU( I ), WORK, LDWORK ) |
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266 * |
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267 * Update the unreduced submatrix A(i+ib:n,i+ib:n), using |
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268 * an update of the form: A := A - V*W' - W*V' |
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269 * |
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270 CALL DSYR2K( UPLO, 'No transpose', N-I-NB+1, NB, -ONE, |
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271 $ A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE, |
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272 $ A( I+NB, I+NB ), LDA ) |
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273 * |
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274 * Copy subdiagonal elements back into A, and diagonal |
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275 * elements into D |
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276 * |
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277 DO 30 J = I, I + NB - 1 |
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278 A( J+1, J ) = E( J ) |
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279 D( J ) = A( J, J ) |
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280 30 CONTINUE |
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281 40 CONTINUE |
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282 * |
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283 * Use unblocked code to reduce the last or only block |
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284 * |
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285 CALL DSYTD2( UPLO, N-I+1, A( I, I ), LDA, D( I ), E( I ), |
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286 $ TAU( I ), IINFO ) |
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287 END IF |
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288 * |
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289 WORK( 1 ) = LWKOPT |
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290 RETURN |
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291 * |
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292 * End of DSYTRD |
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293 * |
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294 END |
