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2329
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1 SUBROUTINE DGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO ) |
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2 * |
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3 * -- LAPACK routine (version 2.0) -- |
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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 * September 30, 1994 |
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7 * |
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8 * .. Scalar Arguments .. |
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9 CHARACTER JOB |
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10 INTEGER IHI, ILO, INFO, LDA, N |
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11 * .. |
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12 * .. Array Arguments .. |
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13 DOUBLE PRECISION A( LDA, * ), SCALE( * ) |
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14 * .. |
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15 * |
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16 * Purpose |
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17 * ======= |
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18 * |
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19 * DGEBAL balances a general real matrix A. This involves, first, |
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20 * permuting A by a similarity transformation to isolate eigenvalues |
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21 * in the first 1 to ILO-1 and last IHI+1 to N elements on the |
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22 * diagonal; and second, applying a diagonal similarity transformation |
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23 * to rows and columns ILO to IHI to make the rows and columns as |
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24 * close in norm as possible. Both steps are optional. |
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25 * |
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26 * Balancing may reduce the 1-norm of the matrix, and improve the |
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27 * accuracy of the computed eigenvalues and/or eigenvectors. |
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28 * |
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29 * Arguments |
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30 * ========= |
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31 * |
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32 * JOB (input) CHARACTER*1 |
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33 * Specifies the operations to be performed on A: |
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34 * = 'N': none: simply set ILO = 1, IHI = N, SCALE(I) = 1.0 |
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35 * for i = 1,...,N; |
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36 * = 'P': permute only; |
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37 * = 'S': scale only; |
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38 * = 'B': both permute and scale. |
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39 * |
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40 * N (input) INTEGER |
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41 * The order of the matrix A. N >= 0. |
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42 * |
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43 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) |
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44 * On entry, the input matrix A. |
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45 * On exit, A is overwritten by the balanced matrix. |
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46 * If JOB = 'N', A is not referenced. |
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47 * See Further Details. |
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48 * |
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49 * LDA (input) INTEGER |
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50 * The leading dimension of the array A. LDA >= max(1,N). |
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51 * |
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52 * ILO (output) INTEGER |
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53 * IHI (output) INTEGER |
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54 * ILO and IHI are set to integers such that on exit |
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55 * A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N. |
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56 * If JOB = 'N' or 'S', ILO = 1 and IHI = N. |
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57 * |
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58 * SCALE (output) DOUBLE PRECISION array, dimension (N) |
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59 * Details of the permutations and scaling factors applied to |
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60 * A. If P(j) is the index of the row and column interchanged |
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61 * with row and column j and D(j) is the scaling factor |
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62 * applied to row and column j, then |
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63 * SCALE(j) = P(j) for j = 1,...,ILO-1 |
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64 * = D(j) for j = ILO,...,IHI |
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65 * = P(j) for j = IHI+1,...,N. |
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66 * The order in which the interchanges are made is N to IHI+1, |
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67 * then 1 to ILO-1. |
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68 * |
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69 * INFO (output) INTEGER |
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70 * = 0: successful exit. |
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71 * < 0: if INFO = -i, the i-th argument had an illegal value. |
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72 * |
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73 * Further Details |
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74 * =============== |
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75 * |
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76 * The permutations consist of row and column interchanges which put |
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77 * the matrix in the form |
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78 * |
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79 * ( T1 X Y ) |
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80 * P A P = ( 0 B Z ) |
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81 * ( 0 0 T2 ) |
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82 * |
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83 * where T1 and T2 are upper triangular matrices whose eigenvalues lie |
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84 * along the diagonal. The column indices ILO and IHI mark the starting |
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85 * and ending columns of the submatrix B. Balancing consists of applying |
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86 * a diagonal similarity transformation inv(D) * B * D to make the |
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87 * 1-norms of each row of B and its corresponding column nearly equal. |
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88 * The output matrix is |
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89 * |
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90 * ( T1 X*D Y ) |
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91 * ( 0 inv(D)*B*D inv(D)*Z ). |
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92 * ( 0 0 T2 ) |
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93 * |
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94 * Information about the permutations P and the diagonal matrix D is |
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95 * returned in the vector SCALE. |
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96 * |
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97 * This subroutine is based on the EISPACK routine BALANC. |
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98 * |
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99 * ===================================================================== |
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100 * |
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101 * .. Parameters .. |
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102 DOUBLE PRECISION ZERO, ONE |
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103 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) |
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104 DOUBLE PRECISION SCLFAC |
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105 PARAMETER ( SCLFAC = 1.0D+1 ) |
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106 DOUBLE PRECISION FACTOR |
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107 PARAMETER ( FACTOR = 0.95D+0 ) |
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108 * .. |
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109 * .. Local Scalars .. |
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110 LOGICAL NOCONV |
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111 INTEGER I, ICA, IEXC, IRA, J, K, L, M |
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112 DOUBLE PRECISION C, CA, F, G, R, RA, S, SFMAX1, SFMAX2, SFMIN1, |
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113 $ SFMIN2 |
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114 * .. |
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115 * .. External Functions .. |
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116 LOGICAL LSAME |
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117 INTEGER IDAMAX |
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118 DOUBLE PRECISION DLAMCH |
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119 EXTERNAL LSAME, IDAMAX, DLAMCH |
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120 * .. |
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121 * .. External Subroutines .. |
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122 EXTERNAL DSCAL, DSWAP, XERBLA |
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123 * .. |
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124 * .. Intrinsic Functions .. |
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125 INTRINSIC ABS, MAX, MIN |
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126 * .. |
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127 * .. Executable Statements .. |
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128 * |
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129 * Test the input parameters |
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130 * |
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131 INFO = 0 |
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132 IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.LSAME( JOB, 'P' ) .AND. |
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133 $ .NOT.LSAME( JOB, 'S' ) .AND. .NOT.LSAME( JOB, 'B' ) ) THEN |
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134 INFO = -1 |
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135 ELSE IF( N.LT.0 ) THEN |
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136 INFO = -2 |
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137 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN |
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138 INFO = -4 |
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139 END IF |
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140 IF( INFO.NE.0 ) THEN |
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141 CALL XERBLA( 'DGEBAL', -INFO ) |
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142 RETURN |
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143 END IF |
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144 * |
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145 K = 1 |
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146 L = N |
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147 * |
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148 IF( N.EQ.0 ) |
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149 $ GO TO 210 |
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150 * |
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151 IF( LSAME( JOB, 'N' ) ) THEN |
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152 DO 10 I = 1, N |
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153 SCALE( I ) = ONE |
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154 10 CONTINUE |
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155 GO TO 210 |
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156 END IF |
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157 * |
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158 IF( LSAME( JOB, 'S' ) ) |
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159 $ GO TO 120 |
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160 * |
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161 * Permutation to isolate eigenvalues if possible |
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162 * |
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163 GO TO 50 |
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164 * |
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165 * Row and column exchange. |
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166 * |
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167 20 CONTINUE |
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168 SCALE( M ) = J |
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169 IF( J.EQ.M ) |
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170 $ GO TO 30 |
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171 * |
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172 CALL DSWAP( L, A( 1, J ), 1, A( 1, M ), 1 ) |
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173 CALL DSWAP( N-K+1, A( J, K ), LDA, A( M, K ), LDA ) |
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174 * |
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175 30 CONTINUE |
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176 GO TO ( 40, 80 )IEXC |
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177 * |
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178 * Search for rows isolating an eigenvalue and push them down. |
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179 * |
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180 40 CONTINUE |
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181 IF( L.EQ.1 ) |
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182 $ GO TO 210 |
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183 L = L - 1 |
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184 * |
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185 50 CONTINUE |
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186 DO 70 J = L, 1, -1 |
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187 * |
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188 DO 60 I = 1, L |
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189 IF( I.EQ.J ) |
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190 $ GO TO 60 |
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191 IF( A( J, I ).NE.ZERO ) |
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192 $ GO TO 70 |
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193 60 CONTINUE |
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194 * |
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195 M = L |
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196 IEXC = 1 |
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197 GO TO 20 |
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198 70 CONTINUE |
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199 * |
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200 GO TO 90 |
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201 * |
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202 * Search for columns isolating an eigenvalue and push them left. |
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203 * |
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204 80 CONTINUE |
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205 K = K + 1 |
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206 * |
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207 90 CONTINUE |
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208 DO 110 J = K, L |
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209 * |
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210 DO 100 I = K, L |
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211 IF( I.EQ.J ) |
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212 $ GO TO 100 |
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213 IF( A( I, J ).NE.ZERO ) |
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214 $ GO TO 110 |
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215 100 CONTINUE |
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216 * |
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217 M = K |
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218 IEXC = 2 |
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219 GO TO 20 |
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220 110 CONTINUE |
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221 * |
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222 120 CONTINUE |
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223 DO 130 I = K, L |
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224 SCALE( I ) = ONE |
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225 130 CONTINUE |
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226 * |
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227 IF( LSAME( JOB, 'P' ) ) |
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228 $ GO TO 210 |
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229 * |
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230 * Balance the submatrix in rows K to L. |
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231 * |
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232 * Iterative loop for norm reduction |
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233 * |
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234 SFMIN1 = DLAMCH( 'S' ) / DLAMCH( 'P' ) |
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235 SFMAX1 = ONE / SFMIN1 |
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236 SFMIN2 = SFMIN1*SCLFAC |
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237 SFMAX2 = ONE / SFMIN2 |
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238 140 CONTINUE |
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239 NOCONV = .FALSE. |
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240 * |
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241 DO 200 I = K, L |
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242 C = ZERO |
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243 R = ZERO |
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244 * |
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245 DO 150 J = K, L |
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246 IF( J.EQ.I ) |
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247 $ GO TO 150 |
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248 C = C + ABS( A( J, I ) ) |
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249 R = R + ABS( A( I, J ) ) |
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250 150 CONTINUE |
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251 ICA = IDAMAX( L, A( 1, I ), 1 ) |
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252 CA = ABS( A( ICA, I ) ) |
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253 IRA = IDAMAX( N-K+1, A( I, K ), LDA ) |
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254 RA = ABS( A( I, IRA+K-1 ) ) |
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255 * |
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256 * Guard against zero C or R due to underflow. |
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257 * |
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258 IF( C.EQ.ZERO .OR. R.EQ.ZERO ) |
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259 $ GO TO 200 |
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260 G = R / SCLFAC |
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261 F = ONE |
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262 S = C + R |
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263 160 CONTINUE |
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264 IF( C.GE.G .OR. MAX( F, C, CA ).GE.SFMAX2 .OR. |
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265 $ MIN( R, G, RA ).LE.SFMIN2 )GO TO 170 |
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266 F = F*SCLFAC |
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267 C = C*SCLFAC |
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268 CA = CA*SCLFAC |
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269 R = R / SCLFAC |
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270 G = G / SCLFAC |
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271 RA = RA / SCLFAC |
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272 GO TO 160 |
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273 * |
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274 170 CONTINUE |
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275 G = C / SCLFAC |
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276 180 CONTINUE |
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277 IF( G.LT.R .OR. MAX( R, RA ).GE.SFMAX2 .OR. |
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278 $ MIN( F, C, G, CA ).LE.SFMIN2 )GO TO 190 |
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279 F = F / SCLFAC |
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280 C = C / SCLFAC |
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281 G = G / SCLFAC |
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282 CA = CA / SCLFAC |
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283 R = R*SCLFAC |
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284 RA = RA*SCLFAC |
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285 GO TO 180 |
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286 * |
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287 * Now balance. |
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288 * |
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289 190 CONTINUE |
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290 IF( ( C+R ).GE.FACTOR*S ) |
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291 $ GO TO 200 |
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292 IF( F.LT.ONE .AND. SCALE( I ).LT.ONE ) THEN |
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293 IF( F*SCALE( I ).LE.SFMIN1 ) |
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294 $ GO TO 200 |
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295 END IF |
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296 IF( F.GT.ONE .AND. SCALE( I ).GT.ONE ) THEN |
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297 IF( SCALE( I ).GE.SFMAX1 / F ) |
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298 $ GO TO 200 |
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299 END IF |
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300 G = ONE / F |
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301 SCALE( I ) = SCALE( I )*F |
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302 NOCONV = .TRUE. |
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303 * |
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304 CALL DSCAL( N-K+1, G, A( I, K ), LDA ) |
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305 CALL DSCAL( L, F, A( 1, I ), 1 ) |
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306 * |
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307 200 CONTINUE |
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308 * |
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309 IF( NOCONV ) |
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310 $ GO TO 140 |
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311 * |
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312 210 CONTINUE |
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313 ILO = K |
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314 IHI = L |
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315 * |
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316 RETURN |
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317 * |
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318 * End of DGEBAL |
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319 * |
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320 END |
