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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 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 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 * Modified by Tzu-Yi Chen, Computer Science Division, University of |
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100 * California at Berkeley, USA |
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101 * |
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102 * ===================================================================== |
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103 * |
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104 * .. Parameters .. |
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105 DOUBLE PRECISION ZERO, ONE |
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106 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) |
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107 DOUBLE PRECISION SCLFAC |
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108 PARAMETER ( SCLFAC = 0.8D+1 ) |
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109 DOUBLE PRECISION FACTOR |
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110 PARAMETER ( FACTOR = 0.95D+0 ) |
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111 * .. |
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112 * .. Local Scalars .. |
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113 LOGICAL NOCONV |
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114 INTEGER I, ICA, IEXC, IRA, J, K, L, M |
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115 DOUBLE PRECISION C, CA, F, G, R, RA, S, SFMAX1, SFMAX2, SFMIN1, |
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116 $ SFMIN2 |
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117 * .. |
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118 * .. External Functions .. |
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119 LOGICAL LSAME |
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120 INTEGER IDAMAX |
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121 DOUBLE PRECISION DLAMCH |
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122 EXTERNAL LSAME, IDAMAX, DLAMCH |
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123 * .. |
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124 * .. External Subroutines .. |
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125 EXTERNAL DSCAL, DSWAP, XERBLA |
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126 * .. |
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127 * .. Intrinsic Functions .. |
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128 INTRINSIC ABS, MAX, MIN |
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129 * .. |
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130 * .. Executable Statements .. |
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131 * |
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132 * Test the input parameters |
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133 * |
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134 INFO = 0 |
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135 IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.LSAME( JOB, 'P' ) .AND. |
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136 $ .NOT.LSAME( JOB, 'S' ) .AND. .NOT.LSAME( JOB, 'B' ) ) THEN |
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137 INFO = -1 |
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138 ELSE IF( N.LT.0 ) THEN |
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139 INFO = -2 |
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140 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN |
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141 INFO = -4 |
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142 END IF |
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143 IF( INFO.NE.0 ) THEN |
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144 CALL XERBLA( 'DGEBAL', -INFO ) |
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145 RETURN |
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146 END IF |
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147 * |
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148 K = 1 |
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149 L = N |
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150 * |
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151 IF( N.EQ.0 ) |
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152 $ GO TO 210 |
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153 * |
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154 IF( LSAME( JOB, 'N' ) ) THEN |
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155 DO 10 I = 1, N |
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156 SCALE( I ) = ONE |
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157 10 CONTINUE |
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158 GO TO 210 |
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159 END IF |
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160 * |
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161 IF( LSAME( JOB, 'S' ) ) |
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162 $ GO TO 120 |
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163 * |
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164 * Permutation to isolate eigenvalues if possible |
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165 * |
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166 GO TO 50 |
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167 * |
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168 * Row and column exchange. |
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169 * |
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170 20 CONTINUE |
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171 SCALE( M ) = J |
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172 IF( J.EQ.M ) |
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173 $ GO TO 30 |
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174 * |
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175 CALL DSWAP( L, A( 1, J ), 1, A( 1, M ), 1 ) |
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176 CALL DSWAP( N-K+1, A( J, K ), LDA, A( M, K ), LDA ) |
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177 * |
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178 30 CONTINUE |
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179 GO TO ( 40, 80 )IEXC |
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180 * |
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181 * Search for rows isolating an eigenvalue and push them down. |
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182 * |
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183 40 CONTINUE |
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184 IF( L.EQ.1 ) |
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185 $ GO TO 210 |
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186 L = L - 1 |
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187 * |
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188 50 CONTINUE |
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189 DO 70 J = L, 1, -1 |
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190 * |
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191 DO 60 I = 1, L |
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192 IF( I.EQ.J ) |
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193 $ GO TO 60 |
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194 IF( A( J, I ).NE.ZERO ) |
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195 $ GO TO 70 |
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196 60 CONTINUE |
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197 * |
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198 M = L |
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199 IEXC = 1 |
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200 GO TO 20 |
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201 70 CONTINUE |
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202 * |
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203 GO TO 90 |
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204 * |
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205 * Search for columns isolating an eigenvalue and push them left. |
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206 * |
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207 80 CONTINUE |
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208 K = K + 1 |
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209 * |
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210 90 CONTINUE |
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211 DO 110 J = K, L |
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212 * |
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213 DO 100 I = K, L |
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214 IF( I.EQ.J ) |
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215 $ GO TO 100 |
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216 IF( A( I, J ).NE.ZERO ) |
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217 $ GO TO 110 |
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218 100 CONTINUE |
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219 * |
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220 M = K |
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221 IEXC = 2 |
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222 GO TO 20 |
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223 110 CONTINUE |
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224 * |
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225 120 CONTINUE |
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226 DO 130 I = K, L |
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227 SCALE( I ) = ONE |
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228 130 CONTINUE |
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229 * |
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230 IF( LSAME( JOB, 'P' ) ) |
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231 $ GO TO 210 |
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232 * |
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233 * Balance the submatrix in rows K to L. |
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234 * |
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235 * Iterative loop for norm reduction |
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236 * |
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237 SFMIN1 = DLAMCH( 'S' ) / DLAMCH( 'P' ) |
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238 SFMAX1 = ONE / SFMIN1 |
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239 SFMIN2 = SFMIN1*SCLFAC |
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240 SFMAX2 = ONE / SFMIN2 |
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241 140 CONTINUE |
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242 NOCONV = .FALSE. |
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243 * |
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244 DO 200 I = K, L |
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245 C = ZERO |
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246 R = ZERO |
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247 * |
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248 DO 150 J = K, L |
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249 IF( J.EQ.I ) |
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250 $ GO TO 150 |
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251 C = C + ABS( A( J, I ) ) |
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252 R = R + ABS( A( I, J ) ) |
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253 150 CONTINUE |
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254 ICA = IDAMAX( L, A( 1, I ), 1 ) |
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255 CA = ABS( A( ICA, I ) ) |
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256 IRA = IDAMAX( N-K+1, A( I, K ), LDA ) |
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257 RA = ABS( A( I, IRA+K-1 ) ) |
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258 * |
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259 * Guard against zero C or R due to underflow. |
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260 * |
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261 IF( C.EQ.ZERO .OR. R.EQ.ZERO ) |
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262 $ GO TO 200 |
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263 G = R / SCLFAC |
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264 F = ONE |
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265 S = C + R |
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266 160 CONTINUE |
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267 IF( C.GE.G .OR. MAX( F, C, CA ).GE.SFMAX2 .OR. |
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268 $ MIN( R, G, RA ).LE.SFMIN2 )GO TO 170 |
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269 F = F*SCLFAC |
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270 C = C*SCLFAC |
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271 CA = CA*SCLFAC |
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272 R = R / SCLFAC |
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273 G = G / SCLFAC |
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274 RA = RA / SCLFAC |
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275 GO TO 160 |
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276 * |
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277 170 CONTINUE |
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278 G = C / SCLFAC |
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279 180 CONTINUE |
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280 IF( G.LT.R .OR. MAX( R, RA ).GE.SFMAX2 .OR. |
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281 $ MIN( F, C, G, CA ).LE.SFMIN2 )GO TO 190 |
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282 F = F / SCLFAC |
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283 C = C / SCLFAC |
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284 G = G / SCLFAC |
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285 CA = CA / SCLFAC |
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286 R = R*SCLFAC |
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287 RA = RA*SCLFAC |
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288 GO TO 180 |
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289 * |
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290 * Now balance. |
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291 * |
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292 190 CONTINUE |
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293 IF( ( C+R ).GE.FACTOR*S ) |
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294 $ GO TO 200 |
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295 IF( F.LT.ONE .AND. SCALE( I ).LT.ONE ) THEN |
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296 IF( F*SCALE( I ).LE.SFMIN1 ) |
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297 $ GO TO 200 |
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298 END IF |
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299 IF( F.GT.ONE .AND. SCALE( I ).GT.ONE ) THEN |
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300 IF( SCALE( I ).GE.SFMAX1 / F ) |
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301 $ GO TO 200 |
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302 END IF |
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303 G = ONE / F |
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304 SCALE( I ) = SCALE( I )*F |
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305 NOCONV = .TRUE. |
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306 * |
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307 CALL DSCAL( N-K+1, G, A( I, K ), LDA ) |
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308 CALL DSCAL( L, F, A( 1, I ), 1 ) |
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309 * |
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310 200 CONTINUE |
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311 * |
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312 IF( NOCONV ) |
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313 $ GO TO 140 |
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314 * |
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315 210 CONTINUE |
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316 ILO = K |
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317 IHI = L |
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318 * |
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319 RETURN |
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320 * |
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321 * End of DGEBAL |
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322 * |
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323 END |
