 LAPACK 3.3.1 Linear Algebra PACKage

# dpbsvx.f

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```00001       SUBROUTINE DPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB,
00002      \$                   EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR,
00003      \$                   WORK, IWORK, INFO )
00004 *
00005 *  -- LAPACK driver routine (version 3.3.1) --
00006 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00007 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00008 *  -- April 2011                                                      --
00009 *
00010 *     .. Scalar Arguments ..
00011       CHARACTER          EQUED, FACT, UPLO
00012       INTEGER            INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
00013       DOUBLE PRECISION   RCOND
00014 *     ..
00015 *     .. Array Arguments ..
00016       INTEGER            IWORK( * )
00017       DOUBLE PRECISION   AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
00018      \$                   BERR( * ), FERR( * ), S( * ), WORK( * ),
00019      \$                   X( LDX, * )
00020 *     ..
00021 *
00022 *  Purpose
00023 *  =======
00024 *
00025 *  DPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
00026 *  compute the solution to a real system of linear equations
00027 *     A * X = B,
00028 *  where A is an N-by-N symmetric positive definite band matrix and X
00029 *  and B are N-by-NRHS matrices.
00030 *
00031 *  Error bounds on the solution and a condition estimate are also
00032 *  provided.
00033 *
00034 *  Description
00035 *  ===========
00036 *
00037 *  The following steps are performed:
00038 *
00039 *  1. If FACT = 'E', real scaling factors are computed to equilibrate
00040 *     the system:
00041 *        diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
00042 *     Whether or not the system will be equilibrated depends on the
00043 *     scaling of the matrix A, but if equilibration is used, A is
00044 *     overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
00045 *
00046 *  2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
00047 *     factor the matrix A (after equilibration if FACT = 'E') as
00048 *        A = U**T * U,  if UPLO = 'U', or
00049 *        A = L * L**T,  if UPLO = 'L',
00050 *     where U is an upper triangular band matrix, and L is a lower
00051 *     triangular band matrix.
00052 *
00053 *  3. If the leading i-by-i principal minor is not positive definite,
00054 *     then the routine returns with INFO = i. Otherwise, the factored
00055 *     form of A is used to estimate the condition number of the matrix
00056 *     A.  If the reciprocal of the condition number is less than machine
00057 *     precision, INFO = N+1 is returned as a warning, but the routine
00058 *     still goes on to solve for X and compute error bounds as
00059 *     described below.
00060 *
00061 *  4. The system of equations is solved for X using the factored form
00062 *     of A.
00063 *
00064 *  5. Iterative refinement is applied to improve the computed solution
00065 *     matrix and calculate error bounds and backward error estimates
00066 *     for it.
00067 *
00068 *  6. If equilibration was used, the matrix X is premultiplied by
00069 *     diag(S) so that it solves the original system before
00070 *     equilibration.
00071 *
00072 *  Arguments
00073 *  =========
00074 *
00075 *  FACT    (input) CHARACTER*1
00076 *          Specifies whether or not the factored form of the matrix A is
00077 *          supplied on entry, and if not, whether the matrix A should be
00078 *          equilibrated before it is factored.
00079 *          = 'F':  On entry, AFB contains the factored form of A.
00080 *                  If EQUED = 'Y', the matrix A has been equilibrated
00081 *                  with scaling factors given by S.  AB and AFB will not
00082 *                  be modified.
00083 *          = 'N':  The matrix A will be copied to AFB and factored.
00084 *          = 'E':  The matrix A will be equilibrated if necessary, then
00085 *                  copied to AFB and factored.
00086 *
00087 *  UPLO    (input) CHARACTER*1
00088 *          = 'U':  Upper triangle of A is stored;
00089 *          = 'L':  Lower triangle of A is stored.
00090 *
00091 *  N       (input) INTEGER
00092 *          The number of linear equations, i.e., the order of the
00093 *          matrix A.  N >= 0.
00094 *
00095 *  KD      (input) INTEGER
00096 *          The number of superdiagonals of the matrix A if UPLO = 'U',
00097 *          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
00098 *
00099 *  NRHS    (input) INTEGER
00100 *          The number of right-hand sides, i.e., the number of columns
00101 *          of the matrices B and X.  NRHS >= 0.
00102 *
00103 *  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
00104 *          On entry, the upper or lower triangle of the symmetric band
00105 *          matrix A, stored in the first KD+1 rows of the array, except
00106 *          if FACT = 'F' and EQUED = 'Y', then A must contain the
00107 *          equilibrated matrix diag(S)*A*diag(S).  The j-th column of A
00108 *          is stored in the j-th column of the array AB as follows:
00109 *          if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
00110 *          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(N,j+KD).
00111 *          See below for further details.
00112 *
00113 *          On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
00114 *          diag(S)*A*diag(S).
00115 *
00116 *  LDAB    (input) INTEGER
00117 *          The leading dimension of the array A.  LDAB >= KD+1.
00118 *
00119 *  AFB     (input or output) DOUBLE PRECISION array, dimension (LDAFB,N)
00120 *          If FACT = 'F', then AFB is an input argument and on entry
00121 *          contains the triangular factor U or L from the Cholesky
00122 *          factorization A = U**T*U or A = L*L**T of the band matrix
00123 *          A, in the same storage format as A (see AB).  If EQUED = 'Y',
00124 *          then AFB is the factored form of the equilibrated matrix A.
00125 *
00126 *          If FACT = 'N', then AFB is an output argument and on exit
00127 *          returns the triangular factor U or L from the Cholesky
00128 *          factorization A = U**T*U or A = L*L**T.
00129 *
00130 *          If FACT = 'E', then AFB is an output argument and on exit
00131 *          returns the triangular factor U or L from the Cholesky
00132 *          factorization A = U**T*U or A = L*L**T of the equilibrated
00133 *          matrix A (see the description of A for the form of the
00134 *          equilibrated matrix).
00135 *
00136 *  LDAFB   (input) INTEGER
00137 *          The leading dimension of the array AFB.  LDAFB >= KD+1.
00138 *
00139 *  EQUED   (input or output) CHARACTER*1
00140 *          Specifies the form of equilibration that was done.
00141 *          = 'N':  No equilibration (always true if FACT = 'N').
00142 *          = 'Y':  Equilibration was done, i.e., A has been replaced by
00143 *                  diag(S) * A * diag(S).
00144 *          EQUED is an input argument if FACT = 'F'; otherwise, it is an
00145 *          output argument.
00146 *
00147 *  S       (input or output) DOUBLE PRECISION array, dimension (N)
00148 *          The scale factors for A; not accessed if EQUED = 'N'.  S is
00149 *          an input argument if FACT = 'F'; otherwise, S is an output
00150 *          argument.  If FACT = 'F' and EQUED = 'Y', each element of S
00151 *          must be positive.
00152 *
00153 *  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
00154 *          On entry, the N-by-NRHS right hand side matrix B.
00155 *          On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
00156 *          B is overwritten by diag(S) * B.
00157 *
00158 *  LDB     (input) INTEGER
00159 *          The leading dimension of the array B.  LDB >= max(1,N).
00160 *
00161 *  X       (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
00162 *          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
00163 *          the original system of equations.  Note that if EQUED = 'Y',
00164 *          A and B are modified on exit, and the solution to the
00165 *          equilibrated system is inv(diag(S))*X.
00166 *
00167 *  LDX     (input) INTEGER
00168 *          The leading dimension of the array X.  LDX >= max(1,N).
00169 *
00170 *  RCOND   (output) DOUBLE PRECISION
00171 *          The estimate of the reciprocal condition number of the matrix
00172 *          A after equilibration (if done).  If RCOND is less than the
00173 *          machine precision (in particular, if RCOND = 0), the matrix
00174 *          is singular to working precision.  This condition is
00175 *          indicated by a return code of INFO > 0.
00176 *
00177 *  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
00178 *          The estimated forward error bound for each solution vector
00179 *          X(j) (the j-th column of the solution matrix X).
00180 *          If XTRUE is the true solution corresponding to X(j), FERR(j)
00181 *          is an estimated upper bound for the magnitude of the largest
00182 *          element in (X(j) - XTRUE) divided by the magnitude of the
00183 *          largest element in X(j).  The estimate is as reliable as
00184 *          the estimate for RCOND, and is almost always a slight
00185 *          overestimate of the true error.
00186 *
00187 *  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
00188 *          The componentwise relative backward error of each solution
00189 *          vector X(j) (i.e., the smallest relative change in
00190 *          any element of A or B that makes X(j) an exact solution).
00191 *
00192 *  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)
00193 *
00194 *  IWORK   (workspace) INTEGER array, dimension (N)
00195 *
00196 *  INFO    (output) INTEGER
00197 *          = 0:  successful exit
00198 *          < 0:  if INFO = -i, the i-th argument had an illegal value
00199 *          > 0:  if INFO = i, and i is
00200 *                <= N:  the leading minor of order i of A is
00201 *                       not positive definite, so the factorization
00202 *                       could not be completed, and the solution has not
00203 *                       been computed. RCOND = 0 is returned.
00204 *                = N+1: U is nonsingular, but RCOND is less than machine
00205 *                       precision, meaning that the matrix is singular
00206 *                       to working precision.  Nevertheless, the
00207 *                       solution and error bounds are computed because
00208 *                       there are a number of situations where the
00209 *                       computed solution can be more accurate than the
00210 *                       value of RCOND would suggest.
00211 *
00212 *  Further Details
00213 *  ===============
00214 *
00215 *  The band storage scheme is illustrated by the following example, when
00216 *  N = 6, KD = 2, and UPLO = 'U':
00217 *
00218 *  Two-dimensional storage of the symmetric matrix A:
00219 *
00220 *     a11  a12  a13
00221 *          a22  a23  a24
00222 *               a33  a34  a35
00223 *                    a44  a45  a46
00224 *                         a55  a56
00225 *     (aij=conjg(aji))         a66
00226 *
00227 *  Band storage of the upper triangle of A:
00228 *
00229 *      *    *   a13  a24  a35  a46
00230 *      *   a12  a23  a34  a45  a56
00231 *     a11  a22  a33  a44  a55  a66
00232 *
00233 *  Similarly, if UPLO = 'L' the format of A is as follows:
00234 *
00235 *     a11  a22  a33  a44  a55  a66
00236 *     a21  a32  a43  a54  a65   *
00237 *     a31  a42  a53  a64   *    *
00238 *
00239 *  Array elements marked * are not used by the routine.
00240 *
00241 *  =====================================================================
00242 *
00243 *     .. Parameters ..
00244       DOUBLE PRECISION   ZERO, ONE
00245       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
00246 *     ..
00247 *     .. Local Scalars ..
00248       LOGICAL            EQUIL, NOFACT, RCEQU, UPPER
00249       INTEGER            I, INFEQU, J, J1, J2
00250       DOUBLE PRECISION   AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
00251 *     ..
00252 *     .. External Functions ..
00253       LOGICAL            LSAME
00254       DOUBLE PRECISION   DLAMCH, DLANSB
00255       EXTERNAL           LSAME, DLAMCH, DLANSB
00256 *     ..
00257 *     .. External Subroutines ..
00258       EXTERNAL           DCOPY, DLACPY, DLAQSB, DPBCON, DPBEQU, DPBRFS,
00259      \$                   DPBTRF, DPBTRS, XERBLA
00260 *     ..
00261 *     .. Intrinsic Functions ..
00262       INTRINSIC          MAX, MIN
00263 *     ..
00264 *     .. Executable Statements ..
00265 *
00266       INFO = 0
00267       NOFACT = LSAME( FACT, 'N' )
00268       EQUIL = LSAME( FACT, 'E' )
00269       UPPER = LSAME( UPLO, 'U' )
00270       IF( NOFACT .OR. EQUIL ) THEN
00271          EQUED = 'N'
00272          RCEQU = .FALSE.
00273       ELSE
00274          RCEQU = LSAME( EQUED, 'Y' )
00275          SMLNUM = DLAMCH( 'Safe minimum' )
00276          BIGNUM = ONE / SMLNUM
00277       END IF
00278 *
00279 *     Test the input parameters.
00280 *
00281       IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
00282      \$     THEN
00283          INFO = -1
00284       ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
00285          INFO = -2
00286       ELSE IF( N.LT.0 ) THEN
00287          INFO = -3
00288       ELSE IF( KD.LT.0 ) THEN
00289          INFO = -4
00290       ELSE IF( NRHS.LT.0 ) THEN
00291          INFO = -5
00292       ELSE IF( LDAB.LT.KD+1 ) THEN
00293          INFO = -7
00294       ELSE IF( LDAFB.LT.KD+1 ) THEN
00295          INFO = -9
00296       ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
00297      \$         ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
00298          INFO = -10
00299       ELSE
00300          IF( RCEQU ) THEN
00301             SMIN = BIGNUM
00302             SMAX = ZERO
00303             DO 10 J = 1, N
00304                SMIN = MIN( SMIN, S( J ) )
00305                SMAX = MAX( SMAX, S( J ) )
00306    10       CONTINUE
00307             IF( SMIN.LE.ZERO ) THEN
00308                INFO = -11
00309             ELSE IF( N.GT.0 ) THEN
00310                SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
00311             ELSE
00312                SCOND = ONE
00313             END IF
00314          END IF
00315          IF( INFO.EQ.0 ) THEN
00316             IF( LDB.LT.MAX( 1, N ) ) THEN
00317                INFO = -13
00318             ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
00319                INFO = -15
00320             END IF
00321          END IF
00322       END IF
00323 *
00324       IF( INFO.NE.0 ) THEN
00325          CALL XERBLA( 'DPBSVX', -INFO )
00326          RETURN
00327       END IF
00328 *
00329       IF( EQUIL ) THEN
00330 *
00331 *        Compute row and column scalings to equilibrate the matrix A.
00332 *
00333          CALL DPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFEQU )
00334          IF( INFEQU.EQ.0 ) THEN
00335 *
00336 *           Equilibrate the matrix.
00337 *
00338             CALL DLAQSB( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED )
00339             RCEQU = LSAME( EQUED, 'Y' )
00340          END IF
00341       END IF
00342 *
00343 *     Scale the right-hand side.
00344 *
00345       IF( RCEQU ) THEN
00346          DO 30 J = 1, NRHS
00347             DO 20 I = 1, N
00348                B( I, J ) = S( I )*B( I, J )
00349    20       CONTINUE
00350    30    CONTINUE
00351       END IF
00352 *
00353       IF( NOFACT .OR. EQUIL ) THEN
00354 *
00355 *        Compute the Cholesky factorization A = U**T *U or A = L*L**T.
00356 *
00357          IF( UPPER ) THEN
00358             DO 40 J = 1, N
00359                J1 = MAX( J-KD, 1 )
00360                CALL DCOPY( J-J1+1, AB( KD+1-J+J1, J ), 1,
00361      \$                     AFB( KD+1-J+J1, J ), 1 )
00362    40       CONTINUE
00363          ELSE
00364             DO 50 J = 1, N
00365                J2 = MIN( J+KD, N )
00366                CALL DCOPY( J2-J+1, AB( 1, J ), 1, AFB( 1, J ), 1 )
00367    50       CONTINUE
00368          END IF
00369 *
00370          CALL DPBTRF( UPLO, N, KD, AFB, LDAFB, INFO )
00371 *
00372 *        Return if INFO is non-zero.
00373 *
00374          IF( INFO.GT.0 )THEN
00375             RCOND = ZERO
00376             RETURN
00377          END IF
00378       END IF
00379 *
00380 *     Compute the norm of the matrix A.
00381 *
00382       ANORM = DLANSB( '1', UPLO, N, KD, AB, LDAB, WORK )
00383 *
00384 *     Compute the reciprocal of the condition number of A.
00385 *
00386       CALL DPBCON( UPLO, N, KD, AFB, LDAFB, ANORM, RCOND, WORK, IWORK,
00387      \$             INFO )
00388 *
00389 *     Compute the solution matrix X.
00390 *
00391       CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
00392       CALL DPBTRS( UPLO, N, KD, NRHS, AFB, LDAFB, X, LDX, INFO )
00393 *
00394 *     Use iterative refinement to improve the computed solution and
00395 *     compute error bounds and backward error estimates for it.
00396 *
00397       CALL DPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X,
00398      \$             LDX, FERR, BERR, WORK, IWORK, INFO )
00399 *
00400 *     Transform the solution matrix X to a solution of the original
00401 *     system.
00402 *
00403       IF( RCEQU ) THEN
00404          DO 70 J = 1, NRHS
00405             DO 60 I = 1, N
00406                X( I, J ) = S( I )*X( I, J )
00407    60       CONTINUE
00408    70    CONTINUE
00409          DO 80 J = 1, NRHS
00410             FERR( J ) = FERR( J ) / SCOND
00411    80    CONTINUE
00412       END IF
00413 *
00414 *     Set INFO = N+1 if the matrix is singular to working precision.
00415 *
00416       IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
00417      \$   INFO = N + 1
00418 *
00419       RETURN
00420 *
00421 *     End of DPBSVX
00422 *
00423       END
```