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1 SUBROUTINE ZTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, W, M, S, |
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2 $ SEP, WORK, LWORK, 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, 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 COMPLEX*16 Q( LDQ, * ), T( LDT, * ), W( * ), WORK( * ) |
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17 * .. |
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18 * |
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19 * Purpose |
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20 * ======= |
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21 * |
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22 * ZTRSEN reorders the Schur factorization of a complex matrix |
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23 * A = Q*T*Q**H, so that a selected cluster of eigenvalues appears in |
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24 * the leading positions on the diagonal of the upper triangular matrix |
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25 * T, and the leading columns of Q form an orthonormal basis of the |
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26 * corresponding right invariant subspace. |
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27 * |
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28 * Optionally the routine computes the reciprocal condition numbers of |
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29 * the cluster of eigenvalues and/or the invariant subspace. |
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30 * |
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31 * Arguments |
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32 * ========= |
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33 * |
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34 * JOB (input) CHARACTER*1 |
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35 * Specifies whether condition numbers are required for the |
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36 * cluster of eigenvalues (S) or the invariant subspace (SEP): |
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37 * = 'N': none; |
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38 * = 'E': for eigenvalues only (S); |
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39 * = 'V': for invariant subspace only (SEP); |
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40 * = 'B': for both eigenvalues and invariant subspace (S and |
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41 * SEP). |
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42 * |
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43 * COMPQ (input) CHARACTER*1 |
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44 * = 'V': update the matrix Q of Schur vectors; |
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45 * = 'N': do not update Q. |
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46 * |
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47 * SELECT (input) LOGICAL array, dimension (N) |
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48 * SELECT specifies the eigenvalues in the selected cluster. To |
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49 * select the j-th eigenvalue, SELECT(j) must be set to .TRUE.. |
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50 * |
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51 * N (input) INTEGER |
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52 * The order of the matrix T. N >= 0. |
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53 * |
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54 * T (input/output) COMPLEX*16 array, dimension (LDT,N) |
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55 * On entry, the upper triangular matrix T. |
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56 * On exit, T is overwritten by the reordered matrix T, with the |
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57 * selected eigenvalues as the leading diagonal elements. |
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58 * |
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59 * LDT (input) INTEGER |
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60 * The leading dimension of the array T. LDT >= max(1,N). |
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61 * |
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62 * Q (input/output) COMPLEX*16 array, dimension (LDQ,N) |
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63 * On entry, if COMPQ = 'V', the matrix Q of Schur vectors. |
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64 * On exit, if COMPQ = 'V', Q has been postmultiplied by the |
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65 * unitary transformation matrix which reorders T; the leading M |
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66 * columns of Q form an orthonormal basis for the specified |
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67 * invariant subspace. |
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68 * If COMPQ = 'N', Q is not referenced. |
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69 * |
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70 * LDQ (input) INTEGER |
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71 * The leading dimension of the array Q. |
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72 * LDQ >= 1; and if COMPQ = 'V', LDQ >= N. |
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73 * |
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74 * W (output) COMPLEX*16 array, dimension (N) |
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75 * The reordered eigenvalues of T, in the same order as they |
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76 * appear on the diagonal of T. |
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77 * |
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78 * M (output) INTEGER |
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79 * The dimension of the specified invariant subspace. |
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80 * 0 <= M <= N. |
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81 * |
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82 * S (output) DOUBLE PRECISION |
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83 * If JOB = 'E' or 'B', S is a lower bound on the reciprocal |
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84 * condition number for the selected cluster of eigenvalues. |
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85 * S cannot underestimate the true reciprocal condition number |
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86 * by more than a factor of sqrt(N). If M = 0 or N, S = 1. |
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87 * If JOB = 'N' or 'V', S is not referenced. |
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88 * |
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89 * SEP (output) DOUBLE PRECISION |
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90 * If JOB = 'V' or 'B', SEP is the estimated reciprocal |
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91 * condition number of the specified invariant subspace. If |
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92 * M = 0 or N, SEP = norm(T). |
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93 * If JOB = 'N' or 'E', SEP is not referenced. |
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94 * |
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95 * WORK (workspace/output) COMPLEX*16 array, dimension (LWORK) |
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96 * If JOB = 'N', WORK is not referenced. Otherwise, |
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97 * on exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
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98 * |
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99 * LWORK (input) INTEGER |
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100 * The dimension of the array WORK. |
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101 * If JOB = 'N', LWORK >= 1; |
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102 * if JOB = 'E', LWORK = M*(N-M); |
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103 * if JOB = 'V' or 'B', LWORK >= 2*M*(N-M). |
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104 * |
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105 * If LWORK = -1, then a workspace query is assumed; the routine |
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106 * only calculates the optimal size of the WORK array, returns |
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107 * this value as the first entry of the WORK array, and no error |
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108 * message related to LWORK is issued by XERBLA. |
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109 * |
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110 * INFO (output) INTEGER |
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111 * = 0: successful exit |
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112 * < 0: if INFO = -i, the i-th argument had an illegal value |
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113 * |
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114 * Further Details |
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115 * =============== |
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116 * |
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117 * ZTRSEN first collects the selected eigenvalues by computing a unitary |
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118 * transformation Z to move them to the top left corner of T. In other |
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119 * words, the selected eigenvalues are the eigenvalues of T11 in: |
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120 * |
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121 * Z'*T*Z = ( T11 T12 ) n1 |
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122 * ( 0 T22 ) n2 |
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123 * n1 n2 |
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124 * |
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125 * where N = n1+n2 and Z' means the conjugate transpose of Z. The first |
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126 * n1 columns of Z span the specified invariant subspace of T. |
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127 * |
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128 * If T has been obtained from the Schur factorization of a matrix |
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129 * A = Q*T*Q', then the reordered Schur factorization of A is given by |
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130 * A = (Q*Z)*(Z'*T*Z)*(Q*Z)', and the first n1 columns of Q*Z span the |
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131 * corresponding invariant subspace of A. |
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132 * |
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133 * The reciprocal condition number of the average of the eigenvalues of |
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134 * T11 may be returned in S. S lies between 0 (very badly conditioned) |
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135 * and 1 (very well conditioned). It is computed as follows. First we |
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136 * compute R so that |
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137 * |
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138 * P = ( I R ) n1 |
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139 * ( 0 0 ) n2 |
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140 * n1 n2 |
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141 * |
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142 * is the projector on the invariant subspace associated with T11. |
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143 * R is the solution of the Sylvester equation: |
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144 * |
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145 * T11*R - R*T22 = T12. |
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146 * |
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147 * Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote |
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148 * the two-norm of M. Then S is computed as the lower bound |
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149 * |
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150 * (1 + F-norm(R)**2)**(-1/2) |
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151 * |
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152 * on the reciprocal of 2-norm(P), the true reciprocal condition number. |
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153 * S cannot underestimate 1 / 2-norm(P) by more than a factor of |
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154 * sqrt(N). |
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155 * |
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156 * An approximate error bound for the computed average of the |
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157 * eigenvalues of T11 is |
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158 * |
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159 * EPS * norm(T) / S |
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160 * |
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161 * where EPS is the machine precision. |
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162 * |
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163 * The reciprocal condition number of the right invariant subspace |
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164 * spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP. |
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165 * SEP is defined as the separation of T11 and T22: |
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166 * |
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167 * sep( T11, T22 ) = sigma-min( C ) |
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168 * |
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169 * where sigma-min(C) is the smallest singular value of the |
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170 * n1*n2-by-n1*n2 matrix |
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171 * |
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172 * C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) ) |
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173 * |
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174 * I(m) is an m by m identity matrix, and kprod denotes the Kronecker |
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175 * product. We estimate sigma-min(C) by the reciprocal of an estimate of |
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176 * the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C) |
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177 * cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2). |
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178 * |
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179 * When SEP is small, small changes in T can cause large changes in |
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180 * the invariant subspace. An approximate bound on the maximum angular |
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181 * error in the computed right invariant subspace is |
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182 * |
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183 * EPS * norm(T) / SEP |
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184 * |
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185 * ===================================================================== |
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186 * |
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187 * .. Parameters .. |
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188 DOUBLE PRECISION ZERO, ONE |
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189 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) |
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190 * .. |
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191 * .. Local Scalars .. |
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192 LOGICAL LQUERY, WANTBH, WANTQ, WANTS, WANTSP |
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193 INTEGER IERR, K, KASE, KS, LWMIN, N1, N2, NN |
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194 DOUBLE PRECISION EST, RNORM, SCALE |
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195 * .. |
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196 * .. Local Arrays .. |
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197 DOUBLE PRECISION RWORK( 1 ) |
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198 * .. |
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199 * .. External Functions .. |
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200 LOGICAL LSAME |
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201 DOUBLE PRECISION ZLANGE |
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202 EXTERNAL LSAME, ZLANGE |
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203 * .. |
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204 * .. External Subroutines .. |
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205 EXTERNAL XERBLA, ZLACON, ZLACPY, ZTREXC, ZTRSYL |
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206 * .. |
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207 * .. Intrinsic Functions .. |
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208 INTRINSIC MAX, SQRT |
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209 * .. |
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210 * .. Executable Statements .. |
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211 * |
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212 * Decode and test the input parameters. |
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213 * |
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214 WANTBH = LSAME( JOB, 'B' ) |
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215 WANTS = LSAME( JOB, 'E' ) .OR. WANTBH |
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216 WANTSP = LSAME( JOB, 'V' ) .OR. WANTBH |
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217 WANTQ = LSAME( COMPQ, 'V' ) |
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218 * |
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219 * Set M to the number of selected eigenvalues. |
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220 * |
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221 M = 0 |
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222 DO 10 K = 1, N |
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223 IF( SELECT( K ) ) |
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224 $ M = M + 1 |
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225 10 CONTINUE |
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226 * |
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227 N1 = M |
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228 N2 = N - M |
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229 NN = N1*N2 |
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230 * |
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231 INFO = 0 |
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232 LQUERY = ( LWORK.EQ.-1 ) |
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233 * |
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234 IF( WANTSP ) THEN |
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235 LWMIN = MAX( 1, 2*NN ) |
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236 ELSE IF( LSAME( JOB, 'N' ) ) THEN |
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237 LWMIN = 1 |
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238 ELSE IF( LSAME( JOB, 'E' ) ) THEN |
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239 LWMIN = MAX( 1, NN ) |
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240 END IF |
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241 * |
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242 IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.WANTS .AND. .NOT.WANTSP ) |
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243 $ THEN |
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244 INFO = -1 |
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245 ELSE IF( .NOT.LSAME( COMPQ, 'N' ) .AND. .NOT.WANTQ ) THEN |
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246 INFO = -2 |
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247 ELSE IF( N.LT.0 ) THEN |
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248 INFO = -4 |
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249 ELSE IF( LDT.LT.MAX( 1, N ) ) THEN |
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250 INFO = -6 |
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251 ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN |
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252 INFO = -8 |
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253 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN |
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254 INFO = -14 |
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255 END IF |
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256 * |
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257 IF( INFO.EQ.0 ) THEN |
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258 WORK( 1 ) = LWMIN |
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259 END IF |
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260 * |
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261 IF( INFO.NE.0 ) THEN |
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262 CALL XERBLA( 'ZTRSEN', -INFO ) |
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263 RETURN |
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264 ELSE IF( LQUERY ) THEN |
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265 RETURN |
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266 END IF |
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267 * |
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268 * Quick return if possible |
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269 * |
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270 IF( M.EQ.N .OR. M.EQ.0 ) THEN |
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271 IF( WANTS ) |
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272 $ S = ONE |
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273 IF( WANTSP ) |
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274 $ SEP = ZLANGE( '1', N, N, T, LDT, RWORK ) |
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275 GO TO 40 |
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276 END IF |
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277 * |
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278 * Collect the selected eigenvalues at the top left corner of T. |
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279 * |
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280 KS = 0 |
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281 DO 20 K = 1, N |
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282 IF( SELECT( K ) ) THEN |
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283 KS = KS + 1 |
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284 * |
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285 * Swap the K-th eigenvalue to position KS. |
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286 * |
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287 IF( K.NE.KS ) |
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288 $ CALL ZTREXC( COMPQ, N, T, LDT, Q, LDQ, K, KS, IERR ) |
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289 END IF |
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290 20 CONTINUE |
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291 * |
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292 IF( WANTS ) THEN |
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293 * |
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294 * Solve the Sylvester equation for R: |
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295 * |
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296 * T11*R - R*T22 = scale*T12 |
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297 * |
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298 CALL ZLACPY( 'F', N1, N2, T( 1, N1+1 ), LDT, WORK, N1 ) |
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299 CALL ZTRSYL( 'N', 'N', -1, N1, N2, T, LDT, T( N1+1, N1+1 ), |
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300 $ LDT, WORK, N1, SCALE, IERR ) |
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301 * |
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302 * Estimate the reciprocal of the condition number of the cluster |
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303 * of eigenvalues. |
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304 * |
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305 RNORM = ZLANGE( 'F', N1, N2, WORK, N1, RWORK ) |
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306 IF( RNORM.EQ.ZERO ) THEN |
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307 S = ONE |
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308 ELSE |
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309 S = SCALE / ( SQRT( SCALE*SCALE / RNORM+RNORM )* |
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310 $ SQRT( RNORM ) ) |
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311 END IF |
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312 END IF |
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313 * |
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314 IF( WANTSP ) THEN |
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315 * |
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316 * Estimate sep(T11,T22). |
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317 * |
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318 EST = ZERO |
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319 KASE = 0 |
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320 30 CONTINUE |
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321 CALL ZLACON( NN, WORK( NN+1 ), WORK, EST, KASE ) |
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322 IF( KASE.NE.0 ) THEN |
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323 IF( KASE.EQ.1 ) THEN |
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324 * |
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325 * Solve T11*R - R*T22 = scale*X. |
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326 * |
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327 CALL ZTRSYL( 'N', 'N', -1, N1, N2, T, LDT, |
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328 $ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE, |
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329 $ IERR ) |
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330 ELSE |
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331 * |
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332 * Solve T11'*R - R*T22' = scale*X. |
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333 * |
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334 CALL ZTRSYL( 'C', 'C', -1, N1, N2, T, LDT, |
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335 $ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE, |
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336 $ IERR ) |
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337 END IF |
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338 GO TO 30 |
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339 END IF |
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340 * |
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341 SEP = SCALE / EST |
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342 END IF |
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343 * |
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344 40 CONTINUE |
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345 * |
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346 * Copy reordered eigenvalues to W. |
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347 * |
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348 DO 50 K = 1, N |
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349 W( K ) = T( K, K ) |
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350 50 CONTINUE |
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351 * |
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352 WORK( 1 ) = LWMIN |
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353 * |
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354 RETURN |
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355 * |
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356 * End of ZTRSEN |
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357 * |
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358 END |
