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1 SUBROUTINE ZGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO ) |
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2 * |
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3 * -- LAPACK routine (version 3.0) -- |
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4 * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., |
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5 * Courant Institute, Argonne National Lab, and Rice University |
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6 * June 30, 1999 |
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7 * |
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8 * .. Scalar Arguments .. |
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9 INTEGER IHI, ILO, INFO, LDA, LWORK, N |
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10 * .. |
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11 * .. Array Arguments .. |
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12 COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * ) |
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13 * .. |
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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 * ZGEHRD reduces a complex general matrix A to upper Hessenberg form H |
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19 * by a unitary similarity transformation: Q' * A * Q = H . |
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20 * |
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21 * Arguments |
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22 * ========= |
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23 * |
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24 * N (input) INTEGER |
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25 * The order of the matrix A. N >= 0. |
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26 * |
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27 * ILO (input) INTEGER |
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28 * IHI (input) INTEGER |
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29 * It is assumed that A is already upper triangular in rows |
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30 * and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally |
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31 * set by a previous call to ZGEBAL; otherwise they should be |
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32 * set to 1 and N respectively. See Further Details. |
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33 * 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. |
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34 * |
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35 * A (input/output) COMPLEX*16 array, dimension (LDA,N) |
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36 * On entry, the N-by-N general matrix to be reduced. |
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37 * On exit, the upper triangle and the first subdiagonal of A |
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38 * are overwritten with the upper Hessenberg matrix H, and the |
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39 * elements below the first subdiagonal, with the array TAU, |
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40 * represent the unitary matrix Q as a product of elementary |
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41 * reflectors. See Further Details. |
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42 * |
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43 * LDA (input) INTEGER |
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44 * The leading dimension of the array A. LDA >= max(1,N). |
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45 * |
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46 * TAU (output) COMPLEX*16 array, dimension (N-1) |
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47 * The scalar factors of the elementary reflectors (see Further |
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48 * Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to |
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49 * zero. |
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50 * |
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51 * WORK (workspace/output) COMPLEX*16 array, dimension (LWORK) |
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52 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
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53 * |
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54 * LWORK (input) INTEGER |
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55 * The length of the array WORK. LWORK >= max(1,N). |
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56 * For optimum performance LWORK >= N*NB, where NB is the |
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57 * optimal blocksize. |
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58 * |
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59 * If LWORK = -1, then a workspace query is assumed; the routine |
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60 * only calculates the optimal size of the WORK array, returns |
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61 * this value as the first entry of the WORK array, and no error |
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62 * message related to LWORK is issued by XERBLA. |
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63 * |
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64 * INFO (output) INTEGER |
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65 * = 0: successful exit |
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66 * < 0: if INFO = -i, the i-th argument had an illegal value. |
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67 * |
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68 * Further Details |
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69 * =============== |
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70 * |
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71 * The matrix Q is represented as a product of (ihi-ilo) elementary |
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72 * reflectors |
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73 * |
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74 * Q = H(ilo) H(ilo+1) . . . H(ihi-1). |
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75 * |
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76 * Each H(i) has the form |
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77 * |
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78 * H(i) = I - tau * v * v' |
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79 * |
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80 * where tau is a complex scalar, and v is a complex vector with |
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81 * v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on |
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82 * exit in A(i+2:ihi,i), and tau in TAU(i). |
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83 * |
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84 * The contents of A are illustrated by the following example, with |
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85 * n = 7, ilo = 2 and ihi = 6: |
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86 * |
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87 * on entry, on exit, |
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88 * |
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89 * ( a a a a a a a ) ( a a h h h h a ) |
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90 * ( a a a a a a ) ( a h h h h a ) |
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91 * ( a a a a a a ) ( h h h h h h ) |
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92 * ( a a a a a a ) ( v2 h h h h h ) |
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93 * ( a a a a a a ) ( v2 v3 h h h h ) |
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94 * ( a a a a a a ) ( v2 v3 v4 h h h ) |
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95 * ( a ) ( a ) |
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96 * |
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97 * where a denotes an element of the original matrix A, h denotes a |
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98 * modified element of the upper Hessenberg matrix H, and vi denotes an |
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99 * element of the vector defining H(i). |
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100 * |
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101 * ===================================================================== |
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102 * |
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103 * .. Parameters .. |
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104 INTEGER NBMAX, LDT |
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105 PARAMETER ( NBMAX = 64, LDT = NBMAX+1 ) |
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106 COMPLEX*16 ZERO, ONE |
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107 PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ), |
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108 $ ONE = ( 1.0D+0, 0.0D+0 ) ) |
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109 * .. |
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110 * .. Local Scalars .. |
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111 LOGICAL LQUERY |
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112 INTEGER I, IB, IINFO, IWS, LDWORK, LWKOPT, NB, NBMIN, |
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113 $ NH, NX |
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114 COMPLEX*16 EI |
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115 * .. |
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116 * .. Local Arrays .. |
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117 COMPLEX*16 T( LDT, NBMAX ) |
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118 * .. |
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119 * .. External Subroutines .. |
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120 EXTERNAL XERBLA, ZGEHD2, ZGEMM, ZLAHRD, ZLARFB |
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121 * .. |
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122 * .. Intrinsic Functions .. |
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123 INTRINSIC MAX, MIN |
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124 * .. |
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125 * .. External Functions .. |
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126 INTEGER ILAENV |
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127 EXTERNAL ILAENV |
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128 * .. |
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129 * .. Executable Statements .. |
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130 * |
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131 * Test the input parameters |
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132 * |
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133 INFO = 0 |
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134 NB = MIN( NBMAX, ILAENV( 1, 'ZGEHRD', ' ', N, ILO, IHI, -1 ) ) |
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135 LWKOPT = N*NB |
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136 WORK( 1 ) = LWKOPT |
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137 LQUERY = ( LWORK.EQ.-1 ) |
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138 IF( N.LT.0 ) THEN |
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139 INFO = -1 |
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140 ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN |
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141 INFO = -2 |
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142 ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN |
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143 INFO = -3 |
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144 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN |
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145 INFO = -5 |
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146 ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN |
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147 INFO = -8 |
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148 END IF |
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149 IF( INFO.NE.0 ) THEN |
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150 CALL XERBLA( 'ZGEHRD', -INFO ) |
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151 RETURN |
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152 ELSE IF( LQUERY ) THEN |
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153 RETURN |
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154 END IF |
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155 * |
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156 * Set elements 1:ILO-1 and IHI:N-1 of TAU to zero |
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157 * |
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158 DO 10 I = 1, ILO - 1 |
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159 TAU( I ) = ZERO |
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160 10 CONTINUE |
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161 DO 20 I = MAX( 1, IHI ), N - 1 |
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162 TAU( I ) = ZERO |
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163 20 CONTINUE |
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164 * |
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165 * Quick return if possible |
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166 * |
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167 NH = IHI - ILO + 1 |
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168 IF( NH.LE.1 ) THEN |
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169 WORK( 1 ) = 1 |
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170 RETURN |
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171 END IF |
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172 * |
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173 NBMIN = 2 |
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174 IWS = 1 |
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175 IF( NB.GT.1 .AND. NB.LT.NH ) THEN |
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176 * |
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177 * Determine when to cross over from blocked to unblocked code |
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178 * (last block is always handled by unblocked code). |
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179 * |
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180 NX = MAX( NB, ILAENV( 3, 'ZGEHRD', ' ', N, ILO, IHI, -1 ) ) |
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181 IF( NX.LT.NH ) THEN |
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182 * |
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183 * Determine if workspace is large enough for blocked code. |
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184 * |
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185 IWS = N*NB |
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186 IF( LWORK.LT.IWS ) THEN |
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187 * |
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188 * Not enough workspace to use optimal NB: determine the |
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189 * minimum value of NB, and reduce NB or force use of |
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190 * unblocked code. |
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191 * |
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192 NBMIN = MAX( 2, ILAENV( 2, 'ZGEHRD', ' ', N, ILO, IHI, |
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193 $ -1 ) ) |
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194 IF( LWORK.GE.N*NBMIN ) THEN |
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195 NB = LWORK / N |
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196 ELSE |
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197 NB = 1 |
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198 END IF |
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199 END IF |
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200 END IF |
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201 END IF |
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202 LDWORK = N |
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203 * |
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204 IF( NB.LT.NBMIN .OR. NB.GE.NH ) THEN |
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205 * |
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206 * Use unblocked code below |
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207 * |
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208 I = ILO |
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209 * |
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210 ELSE |
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211 * |
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212 * Use blocked code |
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213 * |
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214 DO 30 I = ILO, IHI - 1 - NX, NB |
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215 IB = MIN( NB, IHI-I ) |
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216 * |
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217 * Reduce columns i:i+ib-1 to Hessenberg form, returning the |
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218 * matrices V and T of the block reflector H = I - V*T*V' |
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219 * which performs the reduction, and also the matrix Y = A*V*T |
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220 * |
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221 CALL ZLAHRD( IHI, I, IB, A( 1, I ), LDA, TAU( I ), T, LDT, |
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222 $ WORK, LDWORK ) |
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223 * |
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224 * Apply the block reflector H to A(1:ihi,i+ib:ihi) from the |
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225 * right, computing A := A - Y * V'. V(i+ib,ib-1) must be set |
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226 * to 1. |
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227 * |
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228 EI = A( I+IB, I+IB-1 ) |
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229 A( I+IB, I+IB-1 ) = ONE |
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230 CALL ZGEMM( 'No transpose', 'Conjugate transpose', IHI, |
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231 $ IHI-I-IB+1, IB, -ONE, WORK, LDWORK, |
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232 $ A( I+IB, I ), LDA, ONE, A( 1, I+IB ), LDA ) |
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233 A( I+IB, I+IB-1 ) = EI |
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234 * |
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235 * Apply the block reflector H to A(i+1:ihi,i+ib:n) from the |
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236 * left |
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237 * |
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238 CALL ZLARFB( 'Left', 'Conjugate transpose', 'Forward', |
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239 $ 'Columnwise', IHI-I, N-I-IB+1, IB, A( I+1, I ), |
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240 $ LDA, T, LDT, A( I+1, I+IB ), LDA, WORK, |
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241 $ LDWORK ) |
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242 30 CONTINUE |
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243 END IF |
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244 * |
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245 * Use unblocked code to reduce the rest of the matrix |
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246 * |
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247 CALL ZGEHD2( N, I, IHI, A, LDA, TAU, WORK, IINFO ) |
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248 WORK( 1 ) = IWS |
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249 * |
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250 RETURN |
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251 * |
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252 * End of ZGEHRD |
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253 * |
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254 END |
