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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b DPBSTF 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download DPBSTF + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpbstf.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbstf.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbstf.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE DPBSTF( UPLO, N, KD, AB, LDAB, INFO ) 00022 * 00023 * .. Scalar Arguments .. 00024 * CHARACTER UPLO 00025 * INTEGER INFO, KD, LDAB, N 00026 * .. 00027 * .. Array Arguments .. 00028 * DOUBLE PRECISION AB( LDAB, * ) 00029 * .. 00030 * 00031 * 00032 *> \par Purpose: 00033 * ============= 00034 *> 00035 *> \verbatim 00036 *> 00037 *> DPBSTF computes a split Cholesky factorization of a real 00038 *> symmetric positive definite band matrix A. 00039 *> 00040 *> This routine is designed to be used in conjunction with DSBGST. 00041 *> 00042 *> The factorization has the form A = S**T*S where S is a band matrix 00043 *> of the same bandwidth as A and the following structure: 00044 *> 00045 *> S = ( U ) 00046 *> ( M L ) 00047 *> 00048 *> where U is upper triangular of order m = (n+kd)/2, and L is lower 00049 *> triangular of order n-m. 00050 *> \endverbatim 00051 * 00052 * Arguments: 00053 * ========== 00054 * 00055 *> \param[in] UPLO 00056 *> \verbatim 00057 *> UPLO is CHARACTER*1 00058 *> = 'U': Upper triangle of A is stored; 00059 *> = 'L': Lower triangle of A is stored. 00060 *> \endverbatim 00061 *> 00062 *> \param[in] N 00063 *> \verbatim 00064 *> N is INTEGER 00065 *> The order of the matrix A. N >= 0. 00066 *> \endverbatim 00067 *> 00068 *> \param[in] KD 00069 *> \verbatim 00070 *> KD is INTEGER 00071 *> The number of superdiagonals of the matrix A if UPLO = 'U', 00072 *> or the number of subdiagonals if UPLO = 'L'. KD >= 0. 00073 *> \endverbatim 00074 *> 00075 *> \param[in,out] AB 00076 *> \verbatim 00077 *> AB is DOUBLE PRECISION array, dimension (LDAB,N) 00078 *> On entry, the upper or lower triangle of the symmetric band 00079 *> matrix A, stored in the first kd+1 rows of the array. The 00080 *> j-th column of A is stored in the j-th column of the array AB 00081 *> as follows: 00082 *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; 00083 *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). 00084 *> 00085 *> On exit, if INFO = 0, the factor S from the split Cholesky 00086 *> factorization A = S**T*S. See Further Details. 00087 *> \endverbatim 00088 *> 00089 *> \param[in] LDAB 00090 *> \verbatim 00091 *> LDAB is INTEGER 00092 *> The leading dimension of the array AB. LDAB >= KD+1. 00093 *> \endverbatim 00094 *> 00095 *> \param[out] INFO 00096 *> \verbatim 00097 *> INFO is INTEGER 00098 *> = 0: successful exit 00099 *> < 0: if INFO = -i, the i-th argument had an illegal value 00100 *> > 0: if INFO = i, the factorization could not be completed, 00101 *> because the updated element a(i,i) was negative; the 00102 *> matrix A is not positive definite. 00103 *> \endverbatim 00104 * 00105 * Authors: 00106 * ======== 00107 * 00108 *> \author Univ. of Tennessee 00109 *> \author Univ. of California Berkeley 00110 *> \author Univ. of Colorado Denver 00111 *> \author NAG Ltd. 00112 * 00113 *> \date November 2011 00114 * 00115 *> \ingroup doubleOTHERcomputational 00116 * 00117 *> \par Further Details: 00118 * ===================== 00119 *> 00120 *> \verbatim 00121 *> 00122 *> The band storage scheme is illustrated by the following example, when 00123 *> N = 7, KD = 2: 00124 *> 00125 *> S = ( s11 s12 s13 ) 00126 *> ( s22 s23 s24 ) 00127 *> ( s33 s34 ) 00128 *> ( s44 ) 00129 *> ( s53 s54 s55 ) 00130 *> ( s64 s65 s66 ) 00131 *> ( s75 s76 s77 ) 00132 *> 00133 *> If UPLO = 'U', the array AB holds: 00134 *> 00135 *> on entry: on exit: 00136 *> 00137 *> * * a13 a24 a35 a46 a57 * * s13 s24 s53 s64 s75 00138 *> * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54 s65 s76 00139 *> a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 00140 *> 00141 *> If UPLO = 'L', the array AB holds: 00142 *> 00143 *> on entry: on exit: 00144 *> 00145 *> a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 00146 *> a21 a32 a43 a54 a65 a76 * s12 s23 s34 s54 s65 s76 * 00147 *> a31 a42 a53 a64 a64 * * s13 s24 s53 s64 s75 * * 00148 *> 00149 *> Array elements marked * are not used by the routine. 00150 *> \endverbatim 00151 *> 00152 * ===================================================================== 00153 SUBROUTINE DPBSTF( UPLO, N, KD, AB, LDAB, INFO ) 00154 * 00155 * -- LAPACK computational routine (version 3.4.0) -- 00156 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00157 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00158 * November 2011 00159 * 00160 * .. Scalar Arguments .. 00161 CHARACTER UPLO 00162 INTEGER INFO, KD, LDAB, N 00163 * .. 00164 * .. Array Arguments .. 00165 DOUBLE PRECISION AB( LDAB, * ) 00166 * .. 00167 * 00168 * ===================================================================== 00169 * 00170 * .. Parameters .. 00171 DOUBLE PRECISION ONE, ZERO 00172 PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) 00173 * .. 00174 * .. Local Scalars .. 00175 LOGICAL UPPER 00176 INTEGER J, KLD, KM, M 00177 DOUBLE PRECISION AJJ 00178 * .. 00179 * .. External Functions .. 00180 LOGICAL LSAME 00181 EXTERNAL LSAME 00182 * .. 00183 * .. External Subroutines .. 00184 EXTERNAL DSCAL, DSYR, XERBLA 00185 * .. 00186 * .. Intrinsic Functions .. 00187 INTRINSIC MAX, MIN, SQRT 00188 * .. 00189 * .. Executable Statements .. 00190 * 00191 * Test the input parameters. 00192 * 00193 INFO = 0 00194 UPPER = LSAME( UPLO, 'U' ) 00195 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 00196 INFO = -1 00197 ELSE IF( N.LT.0 ) THEN 00198 INFO = -2 00199 ELSE IF( KD.LT.0 ) THEN 00200 INFO = -3 00201 ELSE IF( LDAB.LT.KD+1 ) THEN 00202 INFO = -5 00203 END IF 00204 IF( INFO.NE.0 ) THEN 00205 CALL XERBLA( 'DPBSTF', -INFO ) 00206 RETURN 00207 END IF 00208 * 00209 * Quick return if possible 00210 * 00211 IF( N.EQ.0 ) 00212 $ RETURN 00213 * 00214 KLD = MAX( 1, LDAB-1 ) 00215 * 00216 * Set the splitting point m. 00217 * 00218 M = ( N+KD ) / 2 00219 * 00220 IF( UPPER ) THEN 00221 * 00222 * Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m). 00223 * 00224 DO 10 J = N, M + 1, -1 00225 * 00226 * Compute s(j,j) and test for non-positive-definiteness. 00227 * 00228 AJJ = AB( KD+1, J ) 00229 IF( AJJ.LE.ZERO ) 00230 $ GO TO 50 00231 AJJ = SQRT( AJJ ) 00232 AB( KD+1, J ) = AJJ 00233 KM = MIN( J-1, KD ) 00234 * 00235 * Compute elements j-km:j-1 of the j-th column and update the 00236 * the leading submatrix within the band. 00237 * 00238 CALL DSCAL( KM, ONE / AJJ, AB( KD+1-KM, J ), 1 ) 00239 CALL DSYR( 'Upper', KM, -ONE, AB( KD+1-KM, J ), 1, 00240 $ AB( KD+1, J-KM ), KLD ) 00241 10 CONTINUE 00242 * 00243 * Factorize the updated submatrix A(1:m,1:m) as U**T*U. 00244 * 00245 DO 20 J = 1, M 00246 * 00247 * Compute s(j,j) and test for non-positive-definiteness. 00248 * 00249 AJJ = AB( KD+1, J ) 00250 IF( AJJ.LE.ZERO ) 00251 $ GO TO 50 00252 AJJ = SQRT( AJJ ) 00253 AB( KD+1, J ) = AJJ 00254 KM = MIN( KD, M-J ) 00255 * 00256 * Compute elements j+1:j+km of the j-th row and update the 00257 * trailing submatrix within the band. 00258 * 00259 IF( KM.GT.0 ) THEN 00260 CALL DSCAL( KM, ONE / AJJ, AB( KD, J+1 ), KLD ) 00261 CALL DSYR( 'Upper', KM, -ONE, AB( KD, J+1 ), KLD, 00262 $ AB( KD+1, J+1 ), KLD ) 00263 END IF 00264 20 CONTINUE 00265 ELSE 00266 * 00267 * Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m). 00268 * 00269 DO 30 J = N, M + 1, -1 00270 * 00271 * Compute s(j,j) and test for non-positive-definiteness. 00272 * 00273 AJJ = AB( 1, J ) 00274 IF( AJJ.LE.ZERO ) 00275 $ GO TO 50 00276 AJJ = SQRT( AJJ ) 00277 AB( 1, J ) = AJJ 00278 KM = MIN( J-1, KD ) 00279 * 00280 * Compute elements j-km:j-1 of the j-th row and update the 00281 * trailing submatrix within the band. 00282 * 00283 CALL DSCAL( KM, ONE / AJJ, AB( KM+1, J-KM ), KLD ) 00284 CALL DSYR( 'Lower', KM, -ONE, AB( KM+1, J-KM ), KLD, 00285 $ AB( 1, J-KM ), KLD ) 00286 30 CONTINUE 00287 * 00288 * Factorize the updated submatrix A(1:m,1:m) as U**T*U. 00289 * 00290 DO 40 J = 1, M 00291 * 00292 * Compute s(j,j) and test for non-positive-definiteness. 00293 * 00294 AJJ = AB( 1, J ) 00295 IF( AJJ.LE.ZERO ) 00296 $ GO TO 50 00297 AJJ = SQRT( AJJ ) 00298 AB( 1, J ) = AJJ 00299 KM = MIN( KD, M-J ) 00300 * 00301 * Compute elements j+1:j+km of the j-th column and update the 00302 * trailing submatrix within the band. 00303 * 00304 IF( KM.GT.0 ) THEN 00305 CALL DSCAL( KM, ONE / AJJ, AB( 2, J ), 1 ) 00306 CALL DSYR( 'Lower', KM, -ONE, AB( 2, J ), 1, 00307 $ AB( 1, J+1 ), KLD ) 00308 END IF 00309 40 CONTINUE 00310 END IF 00311 RETURN 00312 * 00313 50 CONTINUE 00314 INFO = J 00315 RETURN 00316 * 00317 * End of DPBSTF 00318 * 00319 END