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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 2.0) -- |
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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 * September 30, 1994 |
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7 * |
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8 * .. Scalar Arguments .. |
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9 CHARACTER UPLO |
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10 INTEGER INFO, LDA, LWORK, N |
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11 * .. |
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12 * .. Array Arguments .. |
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13 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ), |
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14 $ WORK( * ) |
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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 * DSYTRD reduces a real symmetric matrix A to real symmetric |
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21 * tridiagonal form T by an orthogonal similarity transformation: |
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22 * Q**T * A * Q = T. |
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23 * |
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24 * Arguments |
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25 * ========= |
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26 * |
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27 * UPLO (input) CHARACTER*1 |
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28 * = 'U': Upper triangle of A is stored; |
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29 * = 'L': Lower triangle of A is stored. |
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30 * |
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31 * N (input) INTEGER |
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32 * The order of the matrix A. N >= 0. |
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33 * |
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34 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) |
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35 * On entry, the symmetric matrix A. If UPLO = 'U', the leading |
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36 * N-by-N upper triangular part of A contains the upper |
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37 * triangular part of the matrix A, and the strictly lower |
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38 * triangular part of A is not referenced. If UPLO = 'L', the |
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39 * leading N-by-N lower triangular part of A contains the lower |
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40 * triangular part of the matrix A, and the strictly upper |
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41 * triangular part of A is not referenced. |
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42 * On exit, if UPLO = 'U', the diagonal and first superdiagonal |
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43 * of A are overwritten by the corresponding elements of the |
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44 * tridiagonal matrix T, and the elements above the first |
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45 * superdiagonal, with the array TAU, represent the orthogonal |
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46 * matrix Q as a product of elementary reflectors; if UPLO |
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47 * = 'L', the diagonal and first subdiagonal of A are over- |
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48 * written by the corresponding elements of the tridiagonal |
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49 * matrix T, and the elements below the first subdiagonal, with |
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50 * the array TAU, represent the orthogonal matrix Q as a product |
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51 * of elementary reflectors. See Further Details. |
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52 * |
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53 * LDA (input) INTEGER |
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54 * The leading dimension of the array A. LDA >= max(1,N). |
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55 * |
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56 * D (output) DOUBLE PRECISION array, dimension (N) |
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57 * The diagonal elements of the tridiagonal matrix T: |
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58 * D(i) = A(i,i). |
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59 * |
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60 * E (output) DOUBLE PRECISION array, dimension (N-1) |
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61 * The off-diagonal elements of the tridiagonal matrix T: |
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62 * E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. |
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63 * |
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64 * TAU (output) DOUBLE PRECISION array, dimension (N-1) |
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65 * The scalar factors of the elementary reflectors (see Further |
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66 * Details). |
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67 * |
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68 * WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) |
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69 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
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70 * |
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71 * LWORK (input) INTEGER |
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72 * The dimension of the array WORK. LWORK >= 1. |
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73 * For optimum performance LWORK >= N*NB, where NB is the |
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74 * optimal blocksize. |
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75 * |
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76 * INFO (output) INTEGER |
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77 * = 0: successful exit |
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78 * < 0: if INFO = -i, the i-th argument had an illegal value |
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79 * |
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80 * Further Details |
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81 * =============== |
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82 * |
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83 * If UPLO = 'U', the matrix Q is represented as a product of elementary |
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84 * reflectors |
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85 * |
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86 * Q = H(n-1) . . . H(2) H(1). |
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87 * |
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88 * Each H(i) has the form |
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89 * |
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90 * H(i) = I - tau * v * v' |
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91 * |
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92 * where tau is a real scalar, and v is a real vector with |
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93 * v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in |
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94 * A(1:i-1,i+1), and tau in TAU(i). |
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95 * |
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96 * If UPLO = 'L', the matrix Q is represented as a product of elementary |
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97 * reflectors |
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98 * |
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99 * Q = H(1) H(2) . . . H(n-1). |
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100 * |
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101 * Each H(i) has the form |
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102 * |
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103 * H(i) = I - tau * v * v' |
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104 * |
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105 * where tau is a real scalar, and v is a real vector with |
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106 * 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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107 * and tau in TAU(i). |
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108 * |
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109 * The contents of A on exit are illustrated by the following examples |
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110 * with n = 5: |
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111 * |
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112 * if UPLO = 'U': if UPLO = 'L': |
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113 * |
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114 * ( d e v2 v3 v4 ) ( d ) |
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115 * ( d e v3 v4 ) ( e d ) |
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116 * ( d e v4 ) ( v1 e d ) |
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117 * ( d e ) ( v1 v2 e d ) |
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118 * ( d ) ( v1 v2 v3 e d ) |
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119 * |
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120 * where d and e denote diagonal and off-diagonal elements of T, and vi |
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121 * denotes an element of the vector defining H(i). |
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122 * |
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123 * ===================================================================== |
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124 * |
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125 * .. Parameters .. |
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126 DOUBLE PRECISION ONE |
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127 PARAMETER ( ONE = 1.0D0 ) |
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128 * .. |
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129 * .. Local Scalars .. |
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130 LOGICAL UPPER |
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131 INTEGER I, IINFO, IWS, J, KK, LDWORK, NB, NBMIN, NX |
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132 * .. |
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133 * .. External Subroutines .. |
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134 EXTERNAL DLATRD, DSYR2K, DSYTD2, XERBLA |
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135 * .. |
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136 * .. Intrinsic Functions .. |
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137 INTRINSIC MAX |
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138 * .. |
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139 * .. External Functions .. |
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140 LOGICAL LSAME |
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141 INTEGER ILAENV |
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142 EXTERNAL LSAME, ILAENV |
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143 * .. |
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144 * .. Executable Statements .. |
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145 * |
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146 * Test the input parameters |
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147 * |
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148 INFO = 0 |
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149 UPPER = LSAME( UPLO, 'U' ) |
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150 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN |
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151 INFO = -1 |
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152 ELSE IF( N.LT.0 ) THEN |
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153 INFO = -2 |
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154 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN |
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155 INFO = -4 |
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156 ELSE IF( LWORK.LT.1 ) THEN |
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157 INFO = -9 |
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158 END IF |
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159 IF( INFO.NE.0 ) THEN |
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160 CALL XERBLA( 'DSYTRD', -INFO ) |
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161 RETURN |
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162 END IF |
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163 * |
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164 * Quick return if possible |
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165 * |
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166 IF( N.EQ.0 ) THEN |
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167 WORK( 1 ) = 1 |
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168 RETURN |
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169 END IF |
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170 * |
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171 * Determine the block size. |
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172 * |
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173 NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 ) |
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174 NX = N |
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175 IWS = 1 |
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176 IF( NB.GT.1 .AND. NB.LT.N ) THEN |
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177 * |
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178 * Determine when to cross over from blocked to unblocked code |
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179 * (last block is always handled by unblocked code). |
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180 * |
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181 NX = MAX( NB, ILAENV( 3, 'DSYTRD', UPLO, N, -1, -1, -1 ) ) |
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182 IF( NX.LT.N ) THEN |
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183 * |
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184 * Determine if workspace is large enough for blocked code. |
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185 * |
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186 LDWORK = N |
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187 IWS = LDWORK*NB |
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188 IF( LWORK.LT.IWS ) THEN |
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189 * |
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190 * Not enough workspace to use optimal NB: determine the |
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191 * minimum value of NB, and reduce NB or force use of |
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192 * unblocked code by setting NX = N. |
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193 * |
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194 NB = MAX( LWORK / LDWORK, 1 ) |
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195 NBMIN = ILAENV( 2, 'DSYTRD', UPLO, N, -1, -1, -1 ) |
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196 IF( NB.LT.NBMIN ) |
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197 $ NX = N |
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198 END IF |
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199 ELSE |
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200 NX = N |
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201 END IF |
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202 ELSE |
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203 NB = 1 |
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204 END IF |
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205 * |
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206 IF( UPPER ) THEN |
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207 * |
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208 * Reduce the upper triangle of A. |
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209 * Columns 1:kk are handled by the unblocked method. |
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210 * |
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211 KK = N - ( ( N-NX+NB-1 ) / NB )*NB |
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212 DO 20 I = N - NB + 1, KK + 1, -NB |
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213 * |
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214 * Reduce columns i:i+nb-1 to tridiagonal form and form the |
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215 * matrix W which is needed to update the unreduced part of |
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216 * the matrix |
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217 * |
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218 CALL DLATRD( UPLO, I+NB-1, NB, A, LDA, E, TAU, WORK, |
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219 $ LDWORK ) |
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220 * |
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221 * Update the unreduced submatrix A(1:i-1,1:i-1), using an |
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222 * update of the form: A := A - V*W' - W*V' |
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223 * |
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224 CALL DSYR2K( UPLO, 'No transpose', I-1, NB, -ONE, A( 1, I ), |
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225 $ LDA, WORK, LDWORK, ONE, A, LDA ) |
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226 * |
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227 * Copy superdiagonal elements back into A, and diagonal |
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228 * elements into D |
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229 * |
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230 DO 10 J = I, I + NB - 1 |
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231 A( J-1, J ) = E( J-1 ) |
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232 D( J ) = A( J, J ) |
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233 10 CONTINUE |
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234 20 CONTINUE |
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235 * |
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236 * Use unblocked code to reduce the last or only block |
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237 * |
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238 CALL DSYTD2( UPLO, KK, A, LDA, D, E, TAU, IINFO ) |
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239 ELSE |
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240 * |
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241 * Reduce the lower triangle of A |
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242 * |
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243 DO 40 I = 1, N - NX, NB |
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244 * |
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245 * Reduce columns i:i+nb-1 to tridiagonal form and form the |
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246 * matrix W which is needed to update the unreduced part of |
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247 * the matrix |
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248 * |
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249 CALL DLATRD( UPLO, N-I+1, NB, A( I, I ), LDA, E( I ), |
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250 $ TAU( I ), WORK, LDWORK ) |
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251 * |
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252 * Update the unreduced submatrix A(i+ib:n,i+ib:n), using |
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253 * an update of the form: A := A - V*W' - W*V' |
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254 * |
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255 CALL DSYR2K( UPLO, 'No transpose', N-I-NB+1, NB, -ONE, |
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256 $ A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE, |
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257 $ A( I+NB, I+NB ), LDA ) |
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258 * |
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259 * Copy subdiagonal elements back into A, and diagonal |
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260 * elements into D |
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261 * |
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262 DO 30 J = I, I + NB - 1 |
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263 A( J+1, J ) = E( J ) |
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264 D( J ) = A( J, J ) |
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265 30 CONTINUE |
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266 40 CONTINUE |
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267 * |
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268 * Use unblocked code to reduce the last or only block |
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269 * |
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270 CALL DSYTD2( UPLO, N-I+1, A( I, I ), LDA, D( I ), E( I ), |
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271 $ TAU( I ), IINFO ) |
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272 END IF |
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273 * |
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274 WORK( 1 ) = IWS |
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275 RETURN |
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276 * |
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277 * End of DSYTRD |
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278 * |
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279 END |
