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1 SUBROUTINE DPBTF2( UPLO, N, KD, AB, LDAB, INFO ) |
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2 * |
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3 * -- LAPACK routine (version 3.1) -- |
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4 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. |
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5 * November 2006 |
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6 * |
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7 * .. Scalar Arguments .. |
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8 CHARACTER UPLO |
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9 INTEGER INFO, KD, LDAB, N |
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10 * .. |
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11 * .. Array Arguments .. |
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12 DOUBLE PRECISION AB( LDAB, * ) |
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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 * DPBTF2 computes the Cholesky factorization of a real symmetric |
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19 * positive definite band matrix A. |
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20 * |
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21 * The factorization has the form |
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22 * A = U' * U , if UPLO = 'U', or |
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23 * A = L * L', if UPLO = 'L', |
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24 * where U is an upper triangular matrix, U' is the transpose of U, and |
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25 * L is lower triangular. |
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26 * |
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27 * This is the unblocked version of the algorithm, calling Level 2 BLAS. |
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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 * UPLO (input) CHARACTER*1 |
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33 * Specifies whether the upper or lower triangular part of the |
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34 * symmetric matrix A is stored: |
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35 * = 'U': Upper triangular |
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36 * = 'L': Lower triangular |
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37 * |
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38 * N (input) INTEGER |
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39 * The order of the matrix A. N >= 0. |
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40 * |
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41 * KD (input) INTEGER |
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42 * The number of super-diagonals of the matrix A if UPLO = 'U', |
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43 * or the number of sub-diagonals if UPLO = 'L'. KD >= 0. |
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44 * |
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45 * AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N) |
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46 * On entry, the upper or lower triangle of the symmetric band |
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47 * matrix A, stored in the first KD+1 rows of the array. The |
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48 * j-th column of A is stored in the j-th column of the array AB |
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49 * as follows: |
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50 * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; |
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51 * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). |
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52 * |
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53 * On exit, if INFO = 0, the triangular factor U or L from the |
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54 * Cholesky factorization A = U'*U or A = L*L' of the band |
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55 * matrix A, in the same storage format as A. |
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56 * |
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57 * LDAB (input) INTEGER |
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58 * The leading dimension of the array AB. LDAB >= KD+1. |
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59 * |
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60 * INFO (output) INTEGER |
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61 * = 0: successful exit |
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62 * < 0: if INFO = -k, the k-th argument had an illegal value |
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63 * > 0: if INFO = k, the leading minor of order k is not |
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64 * positive definite, and the factorization could not be |
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65 * completed. |
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66 * |
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67 * Further Details |
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68 * =============== |
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69 * |
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70 * The band storage scheme is illustrated by the following example, when |
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71 * N = 6, KD = 2, and UPLO = 'U': |
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72 * |
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73 * On entry: On exit: |
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74 * |
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75 * * * a13 a24 a35 a46 * * u13 u24 u35 u46 |
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76 * * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56 |
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77 * a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66 |
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78 * |
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79 * Similarly, if UPLO = 'L' the format of A is as follows: |
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80 * |
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81 * On entry: On exit: |
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82 * |
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83 * a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66 |
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84 * a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 * |
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85 * a31 a42 a53 a64 * * l31 l42 l53 l64 * * |
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86 * |
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87 * Array elements marked * are not used by the routine. |
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88 * |
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89 * ===================================================================== |
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90 * |
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91 * .. Parameters .. |
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92 DOUBLE PRECISION ONE, ZERO |
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93 PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) |
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94 * .. |
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95 * .. Local Scalars .. |
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96 LOGICAL UPPER |
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97 INTEGER J, KLD, KN |
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98 DOUBLE PRECISION AJJ |
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99 * .. |
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100 * .. External Functions .. |
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101 LOGICAL LSAME |
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102 EXTERNAL LSAME |
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103 * .. |
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104 * .. External Subroutines .. |
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105 EXTERNAL DSCAL, DSYR, XERBLA |
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106 * .. |
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107 * .. Intrinsic Functions .. |
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108 INTRINSIC MAX, MIN, SQRT |
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109 * .. |
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110 * .. Executable Statements .. |
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111 * |
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112 * Test the input parameters. |
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113 * |
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114 INFO = 0 |
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115 UPPER = LSAME( UPLO, 'U' ) |
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116 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN |
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117 INFO = -1 |
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118 ELSE IF( N.LT.0 ) THEN |
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119 INFO = -2 |
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120 ELSE IF( KD.LT.0 ) THEN |
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121 INFO = -3 |
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122 ELSE IF( LDAB.LT.KD+1 ) THEN |
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123 INFO = -5 |
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124 END IF |
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125 IF( INFO.NE.0 ) THEN |
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126 CALL XERBLA( 'DPBTF2', -INFO ) |
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127 RETURN |
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128 END IF |
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129 * |
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130 * Quick return if possible |
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131 * |
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132 IF( N.EQ.0 ) |
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133 $ RETURN |
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134 * |
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135 KLD = MAX( 1, LDAB-1 ) |
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136 * |
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137 IF( UPPER ) THEN |
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138 * |
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139 * Compute the Cholesky factorization A = U'*U. |
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140 * |
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141 DO 10 J = 1, N |
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142 * |
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143 * Compute U(J,J) and test for non-positive-definiteness. |
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144 * |
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145 AJJ = AB( KD+1, J ) |
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146 IF( AJJ.LE.ZERO ) |
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147 $ GO TO 30 |
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148 AJJ = SQRT( AJJ ) |
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149 AB( KD+1, J ) = AJJ |
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150 * |
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151 * Compute elements J+1:J+KN of row J and update the |
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152 * trailing submatrix within the band. |
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153 * |
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154 KN = MIN( KD, N-J ) |
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155 IF( KN.GT.0 ) THEN |
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156 CALL DSCAL( KN, ONE / AJJ, AB( KD, J+1 ), KLD ) |
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157 CALL DSYR( 'Upper', KN, -ONE, AB( KD, J+1 ), KLD, |
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158 $ AB( KD+1, J+1 ), KLD ) |
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159 END IF |
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160 10 CONTINUE |
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161 ELSE |
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162 * |
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163 * Compute the Cholesky factorization A = L*L'. |
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164 * |
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165 DO 20 J = 1, N |
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166 * |
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167 * Compute L(J,J) and test for non-positive-definiteness. |
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168 * |
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169 AJJ = AB( 1, J ) |
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170 IF( AJJ.LE.ZERO ) |
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171 $ GO TO 30 |
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172 AJJ = SQRT( AJJ ) |
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173 AB( 1, J ) = AJJ |
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174 * |
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175 * Compute elements J+1:J+KN of column J and update the |
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176 * trailing submatrix within the band. |
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177 * |
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178 KN = MIN( KD, N-J ) |
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179 IF( KN.GT.0 ) THEN |
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180 CALL DSCAL( KN, ONE / AJJ, AB( 2, J ), 1 ) |
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181 CALL DSYR( 'Lower', KN, -ONE, AB( 2, J ), 1, |
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182 $ AB( 1, J+1 ), KLD ) |
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183 END IF |
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184 20 CONTINUE |
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185 END IF |
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186 RETURN |
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187 * |
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188 30 CONTINUE |
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189 INFO = J |
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190 RETURN |
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191 * |
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192 * End of DPBTF2 |
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193 * |
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194 END |
