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1 SUBROUTINE DGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, |
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2 $ LDQ, Z, LDZ, 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 * September 30, 1994 |
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8 * |
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9 * .. Scalar Arguments .. |
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10 CHARACTER COMPQ, COMPZ |
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11 INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N |
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12 * .. |
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13 * .. Array Arguments .. |
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14 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ), |
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15 $ Z( LDZ, * ) |
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16 * .. |
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17 * |
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18 * Purpose |
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19 * ======= |
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20 * |
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21 * DGGHRD reduces a pair of real matrices (A,B) to generalized upper |
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22 * Hessenberg form using orthogonal transformations, where A is a |
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23 * general matrix and B is upper triangular: Q' * A * Z = H and |
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24 * Q' * B * Z = T, where H is upper Hessenberg, T is upper triangular, |
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25 * and Q and Z are orthogonal, and ' means transpose. |
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26 * |
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27 * The orthogonal matrices Q and Z are determined as products of Givens |
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28 * rotations. They may either be formed explicitly, or they may be |
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29 * postmultiplied into input matrices Q1 and Z1, so that |
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30 * |
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31 * Q1 * A * Z1' = (Q1*Q) * H * (Z1*Z)' |
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32 * Q1 * B * Z1' = (Q1*Q) * T * (Z1*Z)' |
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33 * |
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34 * Arguments |
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35 * ========= |
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36 * |
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37 * COMPQ (input) CHARACTER*1 |
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38 * = 'N': do not compute Q; |
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39 * = 'I': Q is initialized to the unit matrix, and the |
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40 * orthogonal matrix Q is returned; |
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41 * = 'V': Q must contain an orthogonal matrix Q1 on entry, |
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42 * and the product Q1*Q is returned. |
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43 * |
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44 * COMPZ (input) CHARACTER*1 |
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45 * = 'N': do not compute Z; |
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46 * = 'I': Z is initialized to the unit matrix, and the |
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47 * orthogonal matrix Z is returned; |
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48 * = 'V': Z must contain an orthogonal matrix Z1 on entry, |
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49 * and the product Z1*Z is returned. |
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50 * |
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51 * N (input) INTEGER |
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52 * The order of the matrices A and B. N >= 0. |
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53 * |
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54 * ILO (input) INTEGER |
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55 * IHI (input) INTEGER |
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56 * It is assumed that A is already upper triangular in rows and |
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57 * columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set |
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58 * by a previous call to DGGBAL; otherwise they should be set |
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59 * to 1 and N respectively. |
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60 * 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. |
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61 * |
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62 * A (input/output) DOUBLE PRECISION array, dimension (LDA, N) |
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63 * On entry, the N-by-N general matrix to be reduced. |
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64 * On exit, the upper triangle and the first subdiagonal of A |
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65 * are overwritten with the upper Hessenberg matrix H, and the |
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66 * rest is set to zero. |
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67 * |
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68 * LDA (input) INTEGER |
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69 * The leading dimension of the array A. LDA >= max(1,N). |
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70 * |
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71 * B (input/output) DOUBLE PRECISION array, dimension (LDB, N) |
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72 * On entry, the N-by-N upper triangular matrix B. |
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73 * On exit, the upper triangular matrix T = Q' B Z. The |
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74 * elements below the diagonal are set to zero. |
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75 * |
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76 * LDB (input) INTEGER |
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77 * The leading dimension of the array B. LDB >= max(1,N). |
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78 * |
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79 * Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N) |
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80 * If COMPQ='N': Q is not referenced. |
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81 * If COMPQ='I': on entry, Q need not be set, and on exit it |
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82 * contains the orthogonal matrix Q, where Q' |
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83 * is the product of the Givens transformations |
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84 * which are applied to A and B on the left. |
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85 * If COMPQ='V': on entry, Q must contain an orthogonal matrix |
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86 * Q1, and on exit this is overwritten by Q1*Q. |
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87 * |
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88 * LDQ (input) INTEGER |
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89 * The leading dimension of the array Q. |
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90 * LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise. |
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91 * |
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92 * Z (input/output) DOUBLE PRECISION array, dimension (LDZ, N) |
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93 * If COMPZ='N': Z is not referenced. |
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94 * If COMPZ='I': on entry, Z need not be set, and on exit it |
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95 * contains the orthogonal matrix Z, which is |
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96 * the product of the Givens transformations |
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97 * which are applied to A and B on the right. |
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98 * If COMPZ='V': on entry, Z must contain an orthogonal matrix |
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99 * Z1, and on exit this is overwritten by Z1*Z. |
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100 * |
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101 * LDZ (input) INTEGER |
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102 * The leading dimension of the array Z. |
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103 * LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise. |
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104 * |
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105 * INFO (output) INTEGER |
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106 * = 0: successful exit. |
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107 * < 0: if INFO = -i, the i-th argument had an illegal value. |
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108 * |
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109 * Further Details |
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110 * =============== |
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111 * |
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112 * This routine reduces A to Hessenberg and B to triangular form by |
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113 * an unblocked reduction, as described in _Matrix_Computations_, |
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114 * by Golub and Van Loan (Johns Hopkins Press.) |
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115 * |
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116 * ===================================================================== |
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117 * |
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118 * .. Parameters .. |
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119 DOUBLE PRECISION ONE, ZERO |
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120 PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) |
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121 * .. |
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122 * .. Local Scalars .. |
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123 LOGICAL ILQ, ILZ |
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124 INTEGER ICOMPQ, ICOMPZ, JCOL, JROW |
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125 DOUBLE PRECISION C, S, TEMP |
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126 * .. |
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127 * .. External Functions .. |
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128 LOGICAL LSAME |
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129 EXTERNAL LSAME |
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130 * .. |
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131 * .. External Subroutines .. |
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132 EXTERNAL DLARTG, DLASET, DROT, XERBLA |
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133 * .. |
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134 * .. Intrinsic Functions .. |
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135 INTRINSIC MAX |
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136 * .. |
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137 * .. Executable Statements .. |
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138 * |
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139 * Decode COMPQ |
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140 * |
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141 IF( LSAME( COMPQ, 'N' ) ) THEN |
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142 ILQ = .FALSE. |
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143 ICOMPQ = 1 |
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144 ELSE IF( LSAME( COMPQ, 'V' ) ) THEN |
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145 ILQ = .TRUE. |
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146 ICOMPQ = 2 |
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147 ELSE IF( LSAME( COMPQ, 'I' ) ) THEN |
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148 ILQ = .TRUE. |
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149 ICOMPQ = 3 |
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150 ELSE |
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151 ICOMPQ = 0 |
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152 END IF |
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153 * |
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154 * Decode COMPZ |
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155 * |
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156 IF( LSAME( COMPZ, 'N' ) ) THEN |
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157 ILZ = .FALSE. |
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158 ICOMPZ = 1 |
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159 ELSE IF( LSAME( COMPZ, 'V' ) ) THEN |
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160 ILZ = .TRUE. |
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161 ICOMPZ = 2 |
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162 ELSE IF( LSAME( COMPZ, 'I' ) ) THEN |
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163 ILZ = .TRUE. |
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164 ICOMPZ = 3 |
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165 ELSE |
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166 ICOMPZ = 0 |
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167 END IF |
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168 * |
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169 * Test the input parameters. |
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170 * |
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171 INFO = 0 |
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172 IF( ICOMPQ.LE.0 ) THEN |
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173 INFO = -1 |
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174 ELSE IF( ICOMPZ.LE.0 ) THEN |
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175 INFO = -2 |
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176 ELSE IF( N.LT.0 ) THEN |
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177 INFO = -3 |
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178 ELSE IF( ILO.LT.1 ) THEN |
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179 INFO = -4 |
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180 ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN |
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181 INFO = -5 |
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182 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN |
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183 INFO = -7 |
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184 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN |
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185 INFO = -9 |
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186 ELSE IF( ( ILQ .AND. LDQ.LT.N ) .OR. LDQ.LT.1 ) THEN |
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187 INFO = -11 |
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188 ELSE IF( ( ILZ .AND. LDZ.LT.N ) .OR. LDZ.LT.1 ) THEN |
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189 INFO = -13 |
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190 END IF |
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191 IF( INFO.NE.0 ) THEN |
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192 CALL XERBLA( 'DGGHRD', -INFO ) |
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193 RETURN |
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194 END IF |
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195 * |
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196 * Initialize Q and Z if desired. |
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197 * |
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198 IF( ICOMPQ.EQ.3 ) |
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199 $ CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ ) |
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200 IF( ICOMPZ.EQ.3 ) |
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201 $ CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ ) |
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202 * |
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203 * Quick return if possible |
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204 * |
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205 IF( N.LE.1 ) |
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206 $ RETURN |
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207 * |
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208 * Zero out lower triangle of B |
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209 * |
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210 DO 20 JCOL = 1, N - 1 |
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211 DO 10 JROW = JCOL + 1, N |
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212 B( JROW, JCOL ) = ZERO |
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213 10 CONTINUE |
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214 20 CONTINUE |
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215 * |
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216 * Reduce A and B |
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217 * |
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218 DO 40 JCOL = ILO, IHI - 2 |
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219 * |
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220 DO 30 JROW = IHI, JCOL + 2, -1 |
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221 * |
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222 * Step 1: rotate rows JROW-1, JROW to kill A(JROW,JCOL) |
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223 * |
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224 TEMP = A( JROW-1, JCOL ) |
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225 CALL DLARTG( TEMP, A( JROW, JCOL ), C, S, |
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226 $ A( JROW-1, JCOL ) ) |
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227 A( JROW, JCOL ) = ZERO |
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228 CALL DROT( N-JCOL, A( JROW-1, JCOL+1 ), LDA, |
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229 $ A( JROW, JCOL+1 ), LDA, C, S ) |
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230 CALL DROT( N+2-JROW, B( JROW-1, JROW-1 ), LDB, |
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231 $ B( JROW, JROW-1 ), LDB, C, S ) |
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232 IF( ILQ ) |
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233 $ CALL DROT( N, Q( 1, JROW-1 ), 1, Q( 1, JROW ), 1, C, S ) |
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234 * |
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235 * Step 2: rotate columns JROW, JROW-1 to kill B(JROW,JROW-1) |
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236 * |
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237 TEMP = B( JROW, JROW ) |
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238 CALL DLARTG( TEMP, B( JROW, JROW-1 ), C, S, |
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239 $ B( JROW, JROW ) ) |
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240 B( JROW, JROW-1 ) = ZERO |
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241 CALL DROT( IHI, A( 1, JROW ), 1, A( 1, JROW-1 ), 1, C, S ) |
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242 CALL DROT( JROW-1, B( 1, JROW ), 1, B( 1, JROW-1 ), 1, C, |
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243 $ S ) |
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244 IF( ILZ ) |
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245 $ CALL DROT( N, Z( 1, JROW ), 1, Z( 1, JROW-1 ), 1, C, S ) |
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246 30 CONTINUE |
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247 40 CONTINUE |
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248 * |
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249 RETURN |
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250 * |
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251 * End of DGGHRD |
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252 * |
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253 END |
