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1 SUBROUTINE ZHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ, |
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2 $ WORK, LWORK, INFO ) |
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3 * |
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4 * -- LAPACK driver routine (version 3.1) -- |
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5 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. |
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6 * November 2006 |
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7 * |
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8 * .. Scalar Arguments .. |
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9 INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N |
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10 CHARACTER COMPZ, JOB |
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11 * .. |
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12 * .. Array Arguments .. |
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13 COMPLEX*16 H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * ) |
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14 * .. |
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15 * Purpose |
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16 * ======= |
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17 * |
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18 * ZHSEQR computes the eigenvalues of a Hessenberg matrix H |
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19 * and, optionally, the matrices T and Z from the Schur decomposition |
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20 * H = Z T Z**H, where T is an upper triangular matrix (the |
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21 * Schur form), and Z is the unitary matrix of Schur vectors. |
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22 * |
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23 * Optionally Z may be postmultiplied into an input unitary |
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24 * matrix Q so that this routine can give the Schur factorization |
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25 * of a matrix A which has been reduced to the Hessenberg form H |
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26 * by the unitary matrix Q: A = Q*H*Q**H = (QZ)*H*(QZ)**H. |
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27 * |
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28 * Arguments |
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29 * ========= |
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30 * |
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31 * JOB (input) CHARACTER*1 |
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32 * = 'E': compute eigenvalues only; |
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33 * = 'S': compute eigenvalues and the Schur form T. |
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34 * |
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35 * COMPZ (input) CHARACTER*1 |
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36 * = 'N': no Schur vectors are computed; |
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37 * = 'I': Z is initialized to the unit matrix and the matrix Z |
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38 * of Schur vectors of H is returned; |
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39 * = 'V': Z must contain an unitary matrix Q on entry, and |
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40 * the product Q*Z is returned. |
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41 * |
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42 * N (input) INTEGER |
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43 * The order of the matrix H. N .GE. 0. |
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44 * |
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45 * ILO (input) INTEGER |
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46 * IHI (input) INTEGER |
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47 * It is assumed that H is already upper triangular in rows |
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48 * and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally |
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49 * set by a previous call to ZGEBAL, and then passed to ZGEHRD |
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50 * when the matrix output by ZGEBAL is reduced to Hessenberg |
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51 * form. Otherwise ILO and IHI should be set to 1 and N |
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52 * respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N. |
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53 * If N = 0, then ILO = 1 and IHI = 0. |
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54 * |
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55 * H (input/output) COMPLEX*16 array, dimension (LDH,N) |
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56 * On entry, the upper Hessenberg matrix H. |
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57 * On exit, if INFO = 0 and JOB = 'S', H contains the upper |
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58 * triangular matrix T from the Schur decomposition (the |
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59 * Schur form). If INFO = 0 and JOB = 'E', the contents of |
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60 * H are unspecified on exit. (The output value of H when |
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61 * INFO.GT.0 is given under the description of INFO below.) |
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62 * |
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63 * Unlike earlier versions of ZHSEQR, this subroutine may |
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64 * explicitly H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1 |
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65 * or j = IHI+1, IHI+2, ... N. |
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66 * |
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67 * LDH (input) INTEGER |
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68 * The leading dimension of the array H. LDH .GE. max(1,N). |
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69 * |
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70 * W (output) COMPLEX*16 array, dimension (N) |
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71 * The computed eigenvalues. If JOB = 'S', the eigenvalues are |
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72 * stored in the same order as on the diagonal of the Schur |
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73 * form returned in H, with W(i) = H(i,i). |
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74 * |
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75 * Z (input/output) COMPLEX*16 array, dimension (LDZ,N) |
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76 * If COMPZ = 'N', Z is not referenced. |
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77 * If COMPZ = 'I', on entry Z need not be set and on exit, |
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78 * if INFO = 0, Z contains the unitary matrix Z of the Schur |
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79 * vectors of H. If COMPZ = 'V', on entry Z must contain an |
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80 * N-by-N matrix Q, which is assumed to be equal to the unit |
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81 * matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit, |
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82 * if INFO = 0, Z contains Q*Z. |
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83 * Normally Q is the unitary matrix generated by ZUNGHR |
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84 * after the call to ZGEHRD which formed the Hessenberg matrix |
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85 * H. (The output value of Z when INFO.GT.0 is given under |
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86 * the description of INFO below.) |
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87 * |
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88 * LDZ (input) INTEGER |
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89 * The leading dimension of the array Z. if COMPZ = 'I' or |
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90 * COMPZ = 'V', then LDZ.GE.MAX(1,N). Otherwize, LDZ.GE.1. |
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91 * |
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92 * WORK (workspace/output) COMPLEX*16 array, dimension (LWORK) |
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93 * On exit, if INFO = 0, WORK(1) returns an estimate of |
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94 * the optimal value for LWORK. |
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95 * |
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96 * LWORK (input) INTEGER |
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97 * The dimension of the array WORK. LWORK .GE. max(1,N) |
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98 * is sufficient, but LWORK typically as large as 6*N may |
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99 * be required for optimal performance. A workspace query |
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100 * to determine the optimal workspace size is recommended. |
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101 * |
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102 * If LWORK = -1, then ZHSEQR does a workspace query. |
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103 * In this case, ZHSEQR checks the input parameters and |
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104 * estimates the optimal workspace size for the given |
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105 * values of N, ILO and IHI. The estimate is returned |
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106 * in WORK(1). No error message related to LWORK is |
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107 * issued by XERBLA. Neither H nor Z are accessed. |
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108 * |
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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 * .LT. 0: if INFO = -i, the i-th argument had an illegal |
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113 * value |
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114 * .GT. 0: if INFO = i, ZHSEQR failed to compute all of |
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115 * the eigenvalues. Elements 1:ilo-1 and i+1:n of WR |
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116 * and WI contain those eigenvalues which have been |
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117 * successfully computed. (Failures are rare.) |
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118 * |
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119 * If INFO .GT. 0 and JOB = 'E', then on exit, the |
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120 * remaining unconverged eigenvalues are the eigen- |
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121 * values of the upper Hessenberg matrix rows and |
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122 * columns ILO through INFO of the final, output |
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123 * value of H. |
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124 * |
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125 * If INFO .GT. 0 and JOB = 'S', then on exit |
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126 * |
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127 * (*) (initial value of H)*U = U*(final value of H) |
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128 * |
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129 * where U is a unitary matrix. The final |
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130 * value of H is upper Hessenberg and triangular in |
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131 * rows and columns INFO+1 through IHI. |
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132 * |
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133 * If INFO .GT. 0 and COMPZ = 'V', then on exit |
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134 * |
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135 * (final value of Z) = (initial value of Z)*U |
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136 * |
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137 * where U is the unitary matrix in (*) (regard- |
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138 * less of the value of JOB.) |
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139 * |
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140 * If INFO .GT. 0 and COMPZ = 'I', then on exit |
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141 * (final value of Z) = U |
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142 * where U is the unitary matrix in (*) (regard- |
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143 * less of the value of JOB.) |
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144 * |
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145 * If INFO .GT. 0 and COMPZ = 'N', then Z is not |
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146 * accessed. |
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147 * |
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148 * ================================================================ |
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149 * Default values supplied by |
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150 * ILAENV(ISPEC,'ZHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK). |
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151 * It is suggested that these defaults be adjusted in order |
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152 * to attain best performance in each particular |
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153 * computational environment. |
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154 * |
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155 * ISPEC=1: The ZLAHQR vs ZLAQR0 crossover point. |
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156 * Default: 75. (Must be at least 11.) |
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157 * |
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158 * ISPEC=2: Recommended deflation window size. |
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159 * This depends on ILO, IHI and NS. NS is the |
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160 * number of simultaneous shifts returned |
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161 * by ILAENV(ISPEC=4). (See ISPEC=4 below.) |
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162 * The default for (IHI-ILO+1).LE.500 is NS. |
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163 * The default for (IHI-ILO+1).GT.500 is 3*NS/2. |
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164 * |
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165 * ISPEC=3: Nibble crossover point. (See ILAENV for |
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166 * details.) Default: 14% of deflation window |
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167 * size. |
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168 * |
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169 * ISPEC=4: Number of simultaneous shifts, NS, in |
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170 * a multi-shift QR iteration. |
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171 * |
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172 * If IHI-ILO+1 is ... |
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173 * |
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174 * greater than ...but less ... the |
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175 * or equal to ... than default is |
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176 * |
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177 * 1 30 NS - 2(+) |
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178 * 30 60 NS - 4(+) |
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179 * 60 150 NS = 10(+) |
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180 * 150 590 NS = ** |
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181 * 590 3000 NS = 64 |
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182 * 3000 6000 NS = 128 |
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183 * 6000 infinity NS = 256 |
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184 * |
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185 * (+) By default some or all matrices of this order |
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186 * are passed to the implicit double shift routine |
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187 * ZLAHQR and NS is ignored. See ISPEC=1 above |
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188 * and comments in IPARM for details. |
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189 * |
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190 * The asterisks (**) indicate an ad-hoc |
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191 * function of N increasing from 10 to 64. |
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192 * |
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193 * ISPEC=5: Select structured matrix multiply. |
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194 * (See ILAENV for details.) Default: 3. |
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195 * |
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196 * ================================================================ |
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197 * Based on contributions by |
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198 * Karen Braman and Ralph Byers, Department of Mathematics, |
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199 * University of Kansas, USA |
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200 * |
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201 * ================================================================ |
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202 * References: |
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203 * K. Braman, R. Byers and R. Mathias, The Multi-Shift QR |
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204 * Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 |
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205 * Performance, SIAM Journal of Matrix Analysis, volume 23, pages |
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206 * 929--947, 2002. |
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207 * |
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208 * K. Braman, R. Byers and R. Mathias, The Multi-Shift QR |
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209 * Algorithm Part II: Aggressive Early Deflation, SIAM Journal |
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210 * of Matrix Analysis, volume 23, pages 948--973, 2002. |
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211 * |
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212 * ================================================================ |
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213 * .. Parameters .. |
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214 * |
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215 * ==== Matrices of order NTINY or smaller must be processed by |
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216 * . ZLAHQR because of insufficient subdiagonal scratch space. |
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217 * . (This is a hard limit.) ==== |
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218 * |
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219 * ==== NL allocates some local workspace to help small matrices |
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220 * . through a rare ZLAHQR failure. NL .GT. NTINY = 11 is |
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221 * . required and NL .LE. NMIN = ILAENV(ISPEC=1,...) is recom- |
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222 * . mended. (The default value of NMIN is 75.) Using NL = 49 |
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223 * . allows up to six simultaneous shifts and a 16-by-16 |
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224 * . deflation window. ==== |
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225 * |
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226 INTEGER NTINY |
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227 PARAMETER ( NTINY = 11 ) |
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228 INTEGER NL |
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229 PARAMETER ( NL = 49 ) |
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230 COMPLEX*16 ZERO, ONE |
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231 PARAMETER ( ZERO = ( 0.0d0, 0.0d0 ), |
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232 $ ONE = ( 1.0d0, 0.0d0 ) ) |
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233 DOUBLE PRECISION RZERO |
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234 PARAMETER ( RZERO = 0.0d0 ) |
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235 * .. |
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236 * .. Local Arrays .. |
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237 COMPLEX*16 HL( NL, NL ), WORKL( NL ) |
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238 * .. |
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239 * .. Local Scalars .. |
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240 INTEGER KBOT, NMIN |
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241 LOGICAL INITZ, LQUERY, WANTT, WANTZ |
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242 * .. |
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243 * .. External Functions .. |
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244 INTEGER ILAENV |
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245 LOGICAL LSAME |
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246 EXTERNAL ILAENV, LSAME |
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247 * .. |
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248 * .. External Subroutines .. |
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249 EXTERNAL XERBLA, ZCOPY, ZLACPY, ZLAHQR, ZLAQR0, ZLASET |
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250 * .. |
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251 * .. Intrinsic Functions .. |
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252 INTRINSIC DBLE, DCMPLX, MAX, MIN |
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253 * .. |
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254 * .. Executable Statements .. |
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255 * |
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256 * ==== Decode and check the input parameters. ==== |
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257 * |
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258 WANTT = LSAME( JOB, 'S' ) |
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259 INITZ = LSAME( COMPZ, 'I' ) |
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260 WANTZ = INITZ .OR. LSAME( COMPZ, 'V' ) |
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261 WORK( 1 ) = DCMPLX( DBLE( MAX( 1, N ) ), RZERO ) |
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262 LQUERY = LWORK.EQ.-1 |
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263 * |
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264 INFO = 0 |
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265 IF( .NOT.LSAME( JOB, 'E' ) .AND. .NOT.WANTT ) THEN |
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266 INFO = -1 |
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267 ELSE IF( .NOT.LSAME( COMPZ, 'N' ) .AND. .NOT.WANTZ ) THEN |
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268 INFO = -2 |
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269 ELSE IF( N.LT.0 ) THEN |
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270 INFO = -3 |
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271 ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN |
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272 INFO = -4 |
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273 ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN |
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274 INFO = -5 |
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275 ELSE IF( LDH.LT.MAX( 1, N ) ) THEN |
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276 INFO = -7 |
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277 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.MAX( 1, N ) ) ) THEN |
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278 INFO = -10 |
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279 ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN |
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280 INFO = -12 |
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281 END IF |
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282 * |
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283 IF( INFO.NE.0 ) THEN |
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284 * |
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285 * ==== Quick return in case of invalid argument. ==== |
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286 * |
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287 CALL XERBLA( 'ZHSEQR', -INFO ) |
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288 RETURN |
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289 * |
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290 ELSE IF( N.EQ.0 ) THEN |
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291 * |
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292 * ==== Quick return in case N = 0; nothing to do. ==== |
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293 * |
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294 RETURN |
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295 * |
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296 ELSE IF( LQUERY ) THEN |
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297 * |
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298 * ==== Quick return in case of a workspace query ==== |
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299 * |
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300 CALL ZLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILO, IHI, Z, |
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301 $ LDZ, WORK, LWORK, INFO ) |
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302 * ==== Ensure reported workspace size is backward-compatible with |
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303 * . previous LAPACK versions. ==== |
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304 WORK( 1 ) = DCMPLX( MAX( DBLE( WORK( 1 ) ), DBLE( MAX( 1, |
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305 $ N ) ) ), RZERO ) |
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306 RETURN |
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307 * |
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308 ELSE |
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309 * |
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310 * ==== copy eigenvalues isolated by ZGEBAL ==== |
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311 * |
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312 IF( ILO.GT.1 ) |
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313 $ CALL ZCOPY( ILO-1, H, LDH+1, W, 1 ) |
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314 IF( IHI.LT.N ) |
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315 $ CALL ZCOPY( N-IHI, H( IHI+1, IHI+1 ), LDH+1, W( IHI+1 ), 1 ) |
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316 * |
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317 * ==== Initialize Z, if requested ==== |
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318 * |
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319 IF( INITZ ) |
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320 $ CALL ZLASET( 'A', N, N, ZERO, ONE, Z, LDZ ) |
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321 * |
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322 * ==== Quick return if possible ==== |
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323 * |
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324 IF( ILO.EQ.IHI ) THEN |
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325 W( ILO ) = H( ILO, ILO ) |
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326 RETURN |
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327 END IF |
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328 * |
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329 * ==== ZLAHQR/ZLAQR0 crossover point ==== |
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330 * |
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331 NMIN = ILAENV( 1, 'ZHSEQR', JOB( : 1 ) // COMPZ( : 1 ), N, ILO, |
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332 $ IHI, LWORK ) |
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333 NMIN = MAX( NTINY, NMIN ) |
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334 * |
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335 * ==== ZLAQR0 for big matrices; ZLAHQR for small ones ==== |
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336 * |
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337 IF( N.GT.NMIN ) THEN |
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338 CALL ZLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILO, IHI, |
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339 $ Z, LDZ, WORK, LWORK, INFO ) |
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340 ELSE |
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341 * |
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342 * ==== Small matrix ==== |
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343 * |
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344 CALL ZLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILO, IHI, |
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345 $ Z, LDZ, INFO ) |
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346 * |
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347 IF( INFO.GT.0 ) THEN |
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348 * |
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349 * ==== A rare ZLAHQR failure! ZLAQR0 sometimes succeeds |
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350 * . when ZLAHQR fails. ==== |
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351 * |
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352 KBOT = INFO |
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353 * |
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354 IF( N.GE.NL ) THEN |
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355 * |
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356 * ==== Larger matrices have enough subdiagonal scratch |
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357 * . space to call ZLAQR0 directly. ==== |
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358 * |
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359 CALL ZLAQR0( WANTT, WANTZ, N, ILO, KBOT, H, LDH, W, |
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360 $ ILO, IHI, Z, LDZ, WORK, LWORK, INFO ) |
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361 * |
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362 ELSE |
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363 * |
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364 * ==== Tiny matrices don't have enough subdiagonal |
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365 * . scratch space to benefit from ZLAQR0. Hence, |
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366 * . tiny matrices must be copied into a larger |
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367 * . array before calling ZLAQR0. ==== |
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368 * |
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369 CALL ZLACPY( 'A', N, N, H, LDH, HL, NL ) |
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370 HL( N+1, N ) = ZERO |
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371 CALL ZLASET( 'A', NL, NL-N, ZERO, ZERO, HL( 1, N+1 ), |
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372 $ NL ) |
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373 CALL ZLAQR0( WANTT, WANTZ, NL, ILO, KBOT, HL, NL, W, |
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374 $ ILO, IHI, Z, LDZ, WORKL, NL, INFO ) |
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375 IF( WANTT .OR. INFO.NE.0 ) |
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376 $ CALL ZLACPY( 'A', N, N, HL, NL, H, LDH ) |
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377 END IF |
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378 END IF |
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379 END IF |
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380 * |
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381 * ==== Clear out the trash, if necessary. ==== |
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382 * |
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383 IF( ( WANTT .OR. INFO.NE.0 ) .AND. N.GT.2 ) |
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384 $ CALL ZLASET( 'L', N-2, N-2, ZERO, ZERO, H( 3, 1 ), LDH ) |
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385 * |
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386 * ==== Ensure reported workspace size is backward-compatible with |
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387 * . previous LAPACK versions. ==== |
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388 * |
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389 WORK( 1 ) = DCMPLX( MAX( DBLE( MAX( 1, N ) ), |
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390 $ DBLE( WORK( 1 ) ) ), RZERO ) |
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391 END IF |
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392 * |
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393 * ==== End of ZHSEQR ==== |
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394 * |
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395 END |
