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1 SUBROUTINE DTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI, |
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2 $ M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO ) |
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3 * |
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4 * -- LAPACK routine (version 3.0) -- |
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5 * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., |
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6 * Courant Institute, Argonne National Lab, and Rice University |
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7 * June 30, 1999 |
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8 * |
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9 * .. Scalar Arguments .. |
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10 CHARACTER COMPQ, JOB |
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11 INTEGER INFO, LDQ, LDT, LIWORK, LWORK, M, N |
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12 DOUBLE PRECISION S, SEP |
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13 * .. |
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14 * .. Array Arguments .. |
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15 LOGICAL SELECT( * ) |
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16 INTEGER IWORK( * ) |
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17 DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ), |
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18 $ WR( * ) |
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19 * .. |
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20 * |
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21 * Purpose |
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22 * ======= |
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23 * |
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24 * DTRSEN reorders the real Schur factorization of a real matrix |
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25 * A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in |
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26 * the leading diagonal blocks of the upper quasi-triangular matrix T, |
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27 * and the leading columns of Q form an orthonormal basis of the |
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28 * corresponding right invariant subspace. |
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29 * |
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30 * Optionally the routine computes the reciprocal condition numbers of |
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31 * the cluster of eigenvalues and/or the invariant subspace. |
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32 * |
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33 * T must be in Schur canonical form (as returned by DHSEQR), that is, |
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34 * block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each |
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35 * 2-by-2 diagonal block has its diagonal elemnts equal and its |
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36 * off-diagonal elements of opposite sign. |
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37 * |
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38 * Arguments |
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39 * ========= |
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40 * |
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41 * JOB (input) CHARACTER*1 |
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42 * Specifies whether condition numbers are required for the |
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43 * cluster of eigenvalues (S) or the invariant subspace (SEP): |
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44 * = 'N': none; |
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45 * = 'E': for eigenvalues only (S); |
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46 * = 'V': for invariant subspace only (SEP); |
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47 * = 'B': for both eigenvalues and invariant subspace (S and |
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48 * SEP). |
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49 * |
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50 * COMPQ (input) CHARACTER*1 |
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51 * = 'V': update the matrix Q of Schur vectors; |
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52 * = 'N': do not update Q. |
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53 * |
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54 * SELECT (input) LOGICAL array, dimension (N) |
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55 * SELECT specifies the eigenvalues in the selected cluster. To |
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56 * select a real eigenvalue w(j), SELECT(j) must be set to |
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57 * .TRUE.. To select a complex conjugate pair of eigenvalues |
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58 * w(j) and w(j+1), corresponding to a 2-by-2 diagonal block, |
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59 * either SELECT(j) or SELECT(j+1) or both must be set to |
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60 * .TRUE.; a complex conjugate pair of eigenvalues must be |
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61 * either both included in the cluster or both excluded. |
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62 * |
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63 * N (input) INTEGER |
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64 * The order of the matrix T. N >= 0. |
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65 * |
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66 * T (input/output) DOUBLE PRECISION array, dimension (LDT,N) |
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67 * On entry, the upper quasi-triangular matrix T, in Schur |
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68 * canonical form. |
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69 * On exit, T is overwritten by the reordered matrix T, again in |
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70 * Schur canonical form, with the selected eigenvalues in the |
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71 * leading diagonal blocks. |
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72 * |
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73 * LDT (input) INTEGER |
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74 * The leading dimension of the array T. LDT >= max(1,N). |
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75 * |
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76 * Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N) |
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77 * On entry, if COMPQ = 'V', the matrix Q of Schur vectors. |
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78 * On exit, if COMPQ = 'V', Q has been postmultiplied by the |
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79 * orthogonal transformation matrix which reorders T; the |
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80 * leading M columns of Q form an orthonormal basis for the |
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81 * specified invariant subspace. |
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82 * If COMPQ = 'N', Q is not referenced. |
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83 * |
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84 * LDQ (input) INTEGER |
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85 * The leading dimension of the array Q. |
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86 * LDQ >= 1; and if COMPQ = 'V', LDQ >= N. |
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87 * |
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88 * WR (output) DOUBLE PRECISION array, dimension (N) |
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89 * WI (output) DOUBLE PRECISION array, dimension (N) |
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90 * The real and imaginary parts, respectively, of the reordered |
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91 * eigenvalues of T. The eigenvalues are stored in the same |
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92 * order as on the diagonal of T, with WR(i) = T(i,i) and, if |
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93 * T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and |
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94 * WI(i+1) = -WI(i). Note that if a complex eigenvalue is |
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95 * sufficiently ill-conditioned, then its value may differ |
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96 * significantly from its value before reordering. |
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97 * |
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98 * M (output) INTEGER |
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99 * The dimension of the specified invariant subspace. |
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100 * 0 < = M <= N. |
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101 * |
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102 * S (output) DOUBLE PRECISION |
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103 * If JOB = 'E' or 'B', S is a lower bound on the reciprocal |
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104 * condition number for the selected cluster of eigenvalues. |
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105 * S cannot underestimate the true reciprocal condition number |
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106 * by more than a factor of sqrt(N). If M = 0 or N, S = 1. |
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107 * If JOB = 'N' or 'V', S is not referenced. |
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108 * |
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109 * SEP (output) DOUBLE PRECISION |
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110 * If JOB = 'V' or 'B', SEP is the estimated reciprocal |
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111 * condition number of the specified invariant subspace. If |
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112 * M = 0 or N, SEP = norm(T). |
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113 * If JOB = 'N' or 'E', SEP is not referenced. |
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114 * |
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115 * WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) |
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116 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
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117 * |
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118 * LWORK (input) INTEGER |
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119 * The dimension of the array WORK. |
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120 * If JOB = 'N', LWORK >= max(1,N); |
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121 * if JOB = 'E', LWORK >= M*(N-M); |
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122 * if JOB = 'V' or 'B', LWORK >= 2*M*(N-M). |
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123 * |
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124 * If LWORK = -1, then a workspace query is assumed; the routine |
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125 * only calculates the optimal size of the WORK array, returns |
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126 * this value as the first entry of the WORK array, and no error |
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127 * message related to LWORK is issued by XERBLA. |
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128 * |
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129 * IWORK (workspace) INTEGER array, dimension (LIWORK) |
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130 * IF JOB = 'N' or 'E', IWORK is not referenced. |
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131 * |
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132 * LIWORK (input) INTEGER |
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133 * The dimension of the array IWORK. |
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134 * If JOB = 'N' or 'E', LIWORK >= 1; |
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135 * if JOB = 'V' or 'B', LIWORK >= M*(N-M). |
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136 * |
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137 * If LIWORK = -1, then a workspace query is assumed; the |
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138 * routine only calculates the optimal size of the IWORK array, |
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139 * returns this value as the first entry of the IWORK array, and |
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140 * no error message related to LIWORK is issued by XERBLA. |
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141 * |
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142 * INFO (output) INTEGER |
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143 * = 0: successful exit |
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144 * < 0: if INFO = -i, the i-th argument had an illegal value |
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145 * = 1: reordering of T failed because some eigenvalues are too |
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146 * close to separate (the problem is very ill-conditioned); |
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147 * T may have been partially reordered, and WR and WI |
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148 * contain the eigenvalues in the same order as in T; S and |
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149 * SEP (if requested) are set to zero. |
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150 * |
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151 * Further Details |
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152 * =============== |
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153 * |
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154 * DTRSEN first collects the selected eigenvalues by computing an |
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155 * orthogonal transformation Z to move them to the top left corner of T. |
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156 * In other words, the selected eigenvalues are the eigenvalues of T11 |
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157 * in: |
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158 * |
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159 * Z'*T*Z = ( T11 T12 ) n1 |
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160 * ( 0 T22 ) n2 |
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161 * n1 n2 |
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162 * |
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163 * where N = n1+n2 and Z' means the transpose of Z. The first n1 columns |
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164 * of Z span the specified invariant subspace of T. |
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165 * |
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166 * If T has been obtained from the real Schur factorization of a matrix |
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167 * A = Q*T*Q', then the reordered real Schur factorization of A is given |
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168 * by A = (Q*Z)*(Z'*T*Z)*(Q*Z)', and the first n1 columns of Q*Z span |
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169 * the corresponding invariant subspace of A. |
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170 * |
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171 * The reciprocal condition number of the average of the eigenvalues of |
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172 * T11 may be returned in S. S lies between 0 (very badly conditioned) |
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173 * and 1 (very well conditioned). It is computed as follows. First we |
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174 * compute R so that |
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175 * |
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176 * P = ( I R ) n1 |
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177 * ( 0 0 ) n2 |
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178 * n1 n2 |
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179 * |
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180 * is the projector on the invariant subspace associated with T11. |
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181 * R is the solution of the Sylvester equation: |
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182 * |
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183 * T11*R - R*T22 = T12. |
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184 * |
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185 * Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote |
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186 * the two-norm of M. Then S is computed as the lower bound |
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187 * |
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188 * (1 + F-norm(R)**2)**(-1/2) |
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189 * |
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190 * on the reciprocal of 2-norm(P), the true reciprocal condition number. |
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191 * S cannot underestimate 1 / 2-norm(P) by more than a factor of |
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192 * sqrt(N). |
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193 * |
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194 * An approximate error bound for the computed average of the |
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195 * eigenvalues of T11 is |
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196 * |
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197 * EPS * norm(T) / S |
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198 * |
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199 * where EPS is the machine precision. |
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200 * |
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201 * The reciprocal condition number of the right invariant subspace |
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202 * spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP. |
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203 * SEP is defined as the separation of T11 and T22: |
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204 * |
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205 * sep( T11, T22 ) = sigma-min( C ) |
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206 * |
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207 * where sigma-min(C) is the smallest singular value of the |
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208 * n1*n2-by-n1*n2 matrix |
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209 * |
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210 * C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) ) |
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211 * |
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212 * I(m) is an m by m identity matrix, and kprod denotes the Kronecker |
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213 * product. We estimate sigma-min(C) by the reciprocal of an estimate of |
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214 * the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C) |
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215 * cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2). |
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216 * |
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217 * When SEP is small, small changes in T can cause large changes in |
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218 * the invariant subspace. An approximate bound on the maximum angular |
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219 * error in the computed right invariant subspace is |
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220 * |
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221 * EPS * norm(T) / SEP |
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222 * |
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223 * ===================================================================== |
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224 * |
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225 * .. Parameters .. |
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226 DOUBLE PRECISION ZERO, ONE |
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227 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) |
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228 * .. |
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229 * .. Local Scalars .. |
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230 LOGICAL LQUERY, PAIR, SWAP, WANTBH, WANTQ, WANTS, |
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231 $ WANTSP |
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232 INTEGER IERR, K, KASE, KK, KS, LIWMIN, LWMIN, N1, N2, |
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233 $ NN |
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234 DOUBLE PRECISION EST, RNORM, SCALE |
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235 * .. |
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236 * .. External Functions .. |
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237 LOGICAL LSAME |
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238 DOUBLE PRECISION DLANGE |
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239 EXTERNAL LSAME, DLANGE |
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240 * .. |
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241 * .. External Subroutines .. |
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242 EXTERNAL DLACON, DLACPY, DTREXC, DTRSYL, XERBLA |
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243 * .. |
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244 * .. Intrinsic Functions .. |
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245 INTRINSIC ABS, MAX, SQRT |
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246 * .. |
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247 * .. Executable Statements .. |
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248 * |
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249 * Decode and test the input parameters |
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250 * |
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251 WANTBH = LSAME( JOB, 'B' ) |
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252 WANTS = LSAME( JOB, 'E' ) .OR. WANTBH |
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253 WANTSP = LSAME( JOB, 'V' ) .OR. WANTBH |
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254 WANTQ = LSAME( COMPQ, 'V' ) |
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255 * |
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256 INFO = 0 |
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257 LQUERY = ( LWORK.EQ.-1 ) |
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258 IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.WANTS .AND. .NOT.WANTSP ) |
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259 $ THEN |
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260 INFO = -1 |
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261 ELSE IF( .NOT.LSAME( COMPQ, 'N' ) .AND. .NOT.WANTQ ) THEN |
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262 INFO = -2 |
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263 ELSE IF( N.LT.0 ) THEN |
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264 INFO = -4 |
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265 ELSE IF( LDT.LT.MAX( 1, N ) ) THEN |
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266 INFO = -6 |
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267 ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN |
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268 INFO = -8 |
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269 ELSE |
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270 * |
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271 * Set M to the dimension of the specified invariant subspace, |
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272 * and test LWORK and LIWORK. |
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273 * |
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274 M = 0 |
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275 PAIR = .FALSE. |
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276 DO 10 K = 1, N |
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277 IF( PAIR ) THEN |
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278 PAIR = .FALSE. |
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279 ELSE |
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280 IF( K.LT.N ) THEN |
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281 IF( T( K+1, K ).EQ.ZERO ) THEN |
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282 IF( SELECT( K ) ) |
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283 $ M = M + 1 |
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284 ELSE |
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285 PAIR = .TRUE. |
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286 IF( SELECT( K ) .OR. SELECT( K+1 ) ) |
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287 $ M = M + 2 |
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288 END IF |
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289 ELSE |
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290 IF( SELECT( N ) ) |
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291 $ M = M + 1 |
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292 END IF |
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293 END IF |
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294 10 CONTINUE |
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295 * |
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296 N1 = M |
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297 N2 = N - M |
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298 NN = N1*N2 |
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299 * |
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300 IF( WANTSP ) THEN |
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301 LWMIN = MAX( 1, 2*NN ) |
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302 LIWMIN = MAX( 1, NN ) |
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303 ELSE IF( LSAME( JOB, 'N' ) ) THEN |
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304 LWMIN = MAX( 1, N ) |
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305 LIWMIN = 1 |
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306 ELSE IF( LSAME( JOB, 'E' ) ) THEN |
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307 LWMIN = MAX( 1, NN ) |
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308 LIWMIN = 1 |
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309 END IF |
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310 * |
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311 IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN |
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312 INFO = -15 |
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313 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN |
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314 INFO = -17 |
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315 END IF |
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316 END IF |
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317 * |
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318 IF( INFO.EQ.0 ) THEN |
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319 WORK( 1 ) = LWMIN |
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320 IWORK( 1 ) = LIWMIN |
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321 END IF |
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322 * |
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323 IF( INFO.NE.0 ) THEN |
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324 CALL XERBLA( 'DTRSEN', -INFO ) |
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325 RETURN |
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326 ELSE IF( LQUERY ) THEN |
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327 RETURN |
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328 END IF |
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329 * |
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330 * Quick return if possible. |
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331 * |
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332 IF( M.EQ.N .OR. M.EQ.0 ) THEN |
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333 IF( WANTS ) |
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334 $ S = ONE |
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335 IF( WANTSP ) |
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336 $ SEP = DLANGE( '1', N, N, T, LDT, WORK ) |
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337 GO TO 40 |
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338 END IF |
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339 * |
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340 * Collect the selected blocks at the top-left corner of T. |
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341 * |
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342 KS = 0 |
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343 PAIR = .FALSE. |
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344 DO 20 K = 1, N |
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345 IF( PAIR ) THEN |
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346 PAIR = .FALSE. |
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347 ELSE |
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348 SWAP = SELECT( K ) |
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349 IF( K.LT.N ) THEN |
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350 IF( T( K+1, K ).NE.ZERO ) THEN |
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351 PAIR = .TRUE. |
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352 SWAP = SWAP .OR. SELECT( K+1 ) |
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353 END IF |
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354 END IF |
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355 IF( SWAP ) THEN |
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356 KS = KS + 1 |
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357 * |
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358 * Swap the K-th block to position KS. |
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359 * |
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360 IERR = 0 |
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361 KK = K |
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362 IF( K.NE.KS ) |
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363 $ CALL DTREXC( COMPQ, N, T, LDT, Q, LDQ, KK, KS, WORK, |
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364 $ IERR ) |
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365 IF( IERR.EQ.1 .OR. IERR.EQ.2 ) THEN |
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366 * |
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367 * Blocks too close to swap: exit. |
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368 * |
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369 INFO = 1 |
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370 IF( WANTS ) |
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371 $ S = ZERO |
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372 IF( WANTSP ) |
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373 $ SEP = ZERO |
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374 GO TO 40 |
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375 END IF |
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376 IF( PAIR ) |
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377 $ KS = KS + 1 |
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378 END IF |
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379 END IF |
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380 20 CONTINUE |
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381 * |
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382 IF( WANTS ) THEN |
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383 * |
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384 * Solve Sylvester equation for R: |
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385 * |
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386 * T11*R - R*T22 = scale*T12 |
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387 * |
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388 CALL DLACPY( 'F', N1, N2, T( 1, N1+1 ), LDT, WORK, N1 ) |
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389 CALL DTRSYL( 'N', 'N', -1, N1, N2, T, LDT, T( N1+1, N1+1 ), |
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390 $ LDT, WORK, N1, SCALE, IERR ) |
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391 * |
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392 * Estimate the reciprocal of the condition number of the cluster |
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393 * of eigenvalues. |
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394 * |
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395 RNORM = DLANGE( 'F', N1, N2, WORK, N1, WORK ) |
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396 IF( RNORM.EQ.ZERO ) THEN |
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397 S = ONE |
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398 ELSE |
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399 S = SCALE / ( SQRT( SCALE*SCALE / RNORM+RNORM )* |
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400 $ SQRT( RNORM ) ) |
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401 END IF |
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402 END IF |
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403 * |
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404 IF( WANTSP ) THEN |
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405 * |
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406 * Estimate sep(T11,T22). |
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407 * |
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408 EST = ZERO |
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409 KASE = 0 |
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410 30 CONTINUE |
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411 CALL DLACON( NN, WORK( NN+1 ), WORK, IWORK, EST, KASE ) |
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412 IF( KASE.NE.0 ) THEN |
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413 IF( KASE.EQ.1 ) THEN |
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414 * |
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415 * Solve T11*R - R*T22 = scale*X. |
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416 * |
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417 CALL DTRSYL( 'N', 'N', -1, N1, N2, T, LDT, |
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418 $ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE, |
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419 $ IERR ) |
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420 ELSE |
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421 * |
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422 * Solve T11'*R - R*T22' = scale*X. |
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423 * |
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424 CALL DTRSYL( 'T', 'T', -1, N1, N2, T, LDT, |
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425 $ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE, |
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426 $ IERR ) |
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427 END IF |
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428 GO TO 30 |
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429 END IF |
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430 * |
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431 SEP = SCALE / EST |
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432 END IF |
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433 * |
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434 40 CONTINUE |
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435 * |
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436 * Store the output eigenvalues in WR and WI. |
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437 * |
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438 DO 50 K = 1, N |
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439 WR( K ) = T( K, K ) |
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440 WI( K ) = ZERO |
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441 50 CONTINUE |
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442 DO 60 K = 1, N - 1 |
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443 IF( T( K+1, K ).NE.ZERO ) THEN |
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444 WI( K ) = SQRT( ABS( T( K, K+1 ) ) )* |
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445 $ SQRT( ABS( T( K+1, K ) ) ) |
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446 WI( K+1 ) = -WI( K ) |
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447 END IF |
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448 60 CONTINUE |
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3333
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449 * |
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450 WORK( 1 ) = LWMIN |
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451 IWORK( 1 ) = LIWMIN |
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452 * |
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2329
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453 RETURN |
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454 * |
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455 * End of DTRSEN |
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456 * |
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457 END |
