Annotation of rpl/lapack/lapack/dpbsvx.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB,
! 2: $ EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR,
! 3: $ WORK, IWORK, INFO )
! 4: *
! 5: * -- LAPACK driver routine (version 3.2) --
! 6: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 8: * November 2006
! 9: *
! 10: * .. Scalar Arguments ..
! 11: CHARACTER EQUED, FACT, UPLO
! 12: INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
! 13: DOUBLE PRECISION RCOND
! 14: * ..
! 15: * .. Array Arguments ..
! 16: INTEGER IWORK( * )
! 17: DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
! 18: $ BERR( * ), FERR( * ), S( * ), WORK( * ),
! 19: $ X( LDX, * )
! 20: * ..
! 21: *
! 22: * Purpose
! 23: * =======
! 24: *
! 25: * DPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
! 26: * compute the solution to a real system of linear equations
! 27: * A * X = B,
! 28: * where A is an N-by-N symmetric positive definite band matrix and X
! 29: * and B are N-by-NRHS matrices.
! 30: *
! 31: * Error bounds on the solution and a condition estimate are also
! 32: * provided.
! 33: *
! 34: * Description
! 35: * ===========
! 36: *
! 37: * The following steps are performed:
! 38: *
! 39: * 1. If FACT = 'E', real scaling factors are computed to equilibrate
! 40: * the system:
! 41: * diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
! 42: * Whether or not the system will be equilibrated depends on the
! 43: * scaling of the matrix A, but if equilibration is used, A is
! 44: * overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
! 45: *
! 46: * 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
! 47: * factor the matrix A (after equilibration if FACT = 'E') as
! 48: * A = U**T * U, if UPLO = 'U', or
! 49: * A = L * L**T, if UPLO = 'L',
! 50: * where U is an upper triangular band matrix, and L is a lower
! 51: * triangular band matrix.
! 52: *
! 53: * 3. If the leading i-by-i principal minor is not positive definite,
! 54: * then the routine returns with INFO = i. Otherwise, the factored
! 55: * form of A is used to estimate the condition number of the matrix
! 56: * A. If the reciprocal of the condition number is less than machine
! 57: * precision, INFO = N+1 is returned as a warning, but the routine
! 58: * still goes on to solve for X and compute error bounds as
! 59: * described below.
! 60: *
! 61: * 4. The system of equations is solved for X using the factored form
! 62: * of A.
! 63: *
! 64: * 5. Iterative refinement is applied to improve the computed solution
! 65: * matrix and calculate error bounds and backward error estimates
! 66: * for it.
! 67: *
! 68: * 6. If equilibration was used, the matrix X is premultiplied by
! 69: * diag(S) so that it solves the original system before
! 70: * equilibration.
! 71: *
! 72: * Arguments
! 73: * =========
! 74: *
! 75: * FACT (input) CHARACTER*1
! 76: * Specifies whether or not the factored form of the matrix A is
! 77: * supplied on entry, and if not, whether the matrix A should be
! 78: * equilibrated before it is factored.
! 79: * = 'F': On entry, AFB contains the factored form of A.
! 80: * If EQUED = 'Y', the matrix A has been equilibrated
! 81: * with scaling factors given by S. AB and AFB will not
! 82: * be modified.
! 83: * = 'N': The matrix A will be copied to AFB and factored.
! 84: * = 'E': The matrix A will be equilibrated if necessary, then
! 85: * copied to AFB and factored.
! 86: *
! 87: * UPLO (input) CHARACTER*1
! 88: * = 'U': Upper triangle of A is stored;
! 89: * = 'L': Lower triangle of A is stored.
! 90: *
! 91: * N (input) INTEGER
! 92: * The number of linear equations, i.e., the order of the
! 93: * matrix A. N >= 0.
! 94: *
! 95: * KD (input) INTEGER
! 96: * The number of superdiagonals of the matrix A if UPLO = 'U',
! 97: * or the number of subdiagonals if UPLO = 'L'. KD >= 0.
! 98: *
! 99: * NRHS (input) INTEGER
! 100: * The number of right-hand sides, i.e., the number of columns
! 101: * of the matrices B and X. NRHS >= 0.
! 102: *
! 103: * AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
! 104: * On entry, the upper or lower triangle of the symmetric band
! 105: * matrix A, stored in the first KD+1 rows of the array, except
! 106: * if FACT = 'F' and EQUED = 'Y', then A must contain the
! 107: * equilibrated matrix diag(S)*A*diag(S). The j-th column of A
! 108: * is stored in the j-th column of the array AB as follows:
! 109: * if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
! 110: * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(N,j+KD).
! 111: * See below for further details.
! 112: *
! 113: * On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
! 114: * diag(S)*A*diag(S).
! 115: *
! 116: * LDAB (input) INTEGER
! 117: * The leading dimension of the array A. LDAB >= KD+1.
! 118: *
! 119: * AFB (input or output) DOUBLE PRECISION array, dimension (LDAFB,N)
! 120: * If FACT = 'F', then AFB is an input argument and on entry
! 121: * contains the triangular factor U or L from the Cholesky
! 122: * factorization A = U**T*U or A = L*L**T of the band matrix
! 123: * A, in the same storage format as A (see AB). If EQUED = 'Y',
! 124: * then AFB is the factored form of the equilibrated matrix A.
! 125: *
! 126: * If FACT = 'N', then AFB is an output argument and on exit
! 127: * returns the triangular factor U or L from the Cholesky
! 128: * factorization A = U**T*U or A = L*L**T.
! 129: *
! 130: * If FACT = 'E', then AFB is an output argument and on exit
! 131: * returns the triangular factor U or L from the Cholesky
! 132: * factorization A = U**T*U or A = L*L**T of the equilibrated
! 133: * matrix A (see the description of A for the form of the
! 134: * equilibrated matrix).
! 135: *
! 136: * LDAFB (input) INTEGER
! 137: * The leading dimension of the array AFB. LDAFB >= KD+1.
! 138: *
! 139: * EQUED (input or output) CHARACTER*1
! 140: * Specifies the form of equilibration that was done.
! 141: * = 'N': No equilibration (always true if FACT = 'N').
! 142: * = 'Y': Equilibration was done, i.e., A has been replaced by
! 143: * diag(S) * A * diag(S).
! 144: * EQUED is an input argument if FACT = 'F'; otherwise, it is an
! 145: * output argument.
! 146: *
! 147: * S (input or output) DOUBLE PRECISION array, dimension (N)
! 148: * The scale factors for A; not accessed if EQUED = 'N'. S is
! 149: * an input argument if FACT = 'F'; otherwise, S is an output
! 150: * argument. If FACT = 'F' and EQUED = 'Y', each element of S
! 151: * must be positive.
! 152: *
! 153: * B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
! 154: * On entry, the N-by-NRHS right hand side matrix B.
! 155: * On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
! 156: * B is overwritten by diag(S) * B.
! 157: *
! 158: * LDB (input) INTEGER
! 159: * The leading dimension of the array B. LDB >= max(1,N).
! 160: *
! 161: * X (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
! 162: * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
! 163: * the original system of equations. Note that if EQUED = 'Y',
! 164: * A and B are modified on exit, and the solution to the
! 165: * equilibrated system is inv(diag(S))*X.
! 166: *
! 167: * LDX (input) INTEGER
! 168: * The leading dimension of the array X. LDX >= max(1,N).
! 169: *
! 170: * RCOND (output) DOUBLE PRECISION
! 171: * The estimate of the reciprocal condition number of the matrix
! 172: * A after equilibration (if done). If RCOND is less than the
! 173: * machine precision (in particular, if RCOND = 0), the matrix
! 174: * is singular to working precision. This condition is
! 175: * indicated by a return code of INFO > 0.
! 176: *
! 177: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 178: * The estimated forward error bound for each solution vector
! 179: * X(j) (the j-th column of the solution matrix X).
! 180: * If XTRUE is the true solution corresponding to X(j), FERR(j)
! 181: * is an estimated upper bound for the magnitude of the largest
! 182: * element in (X(j) - XTRUE) divided by the magnitude of the
! 183: * largest element in X(j). The estimate is as reliable as
! 184: * the estimate for RCOND, and is almost always a slight
! 185: * overestimate of the true error.
! 186: *
! 187: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 188: * The componentwise relative backward error of each solution
! 189: * vector X(j) (i.e., the smallest relative change in
! 190: * any element of A or B that makes X(j) an exact solution).
! 191: *
! 192: * WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
! 193: *
! 194: * IWORK (workspace) INTEGER array, dimension (N)
! 195: *
! 196: * INFO (output) INTEGER
! 197: * = 0: successful exit
! 198: * < 0: if INFO = -i, the i-th argument had an illegal value
! 199: * > 0: if INFO = i, and i is
! 200: * <= N: the leading minor of order i of A is
! 201: * not positive definite, so the factorization
! 202: * could not be completed, and the solution has not
! 203: * been computed. RCOND = 0 is returned.
! 204: * = N+1: U is nonsingular, but RCOND is less than machine
! 205: * precision, meaning that the matrix is singular
! 206: * to working precision. Nevertheless, the
! 207: * solution and error bounds are computed because
! 208: * there are a number of situations where the
! 209: * computed solution can be more accurate than the
! 210: * value of RCOND would suggest.
! 211: *
! 212: * Further Details
! 213: * ===============
! 214: *
! 215: * The band storage scheme is illustrated by the following example, when
! 216: * N = 6, KD = 2, and UPLO = 'U':
! 217: *
! 218: * Two-dimensional storage of the symmetric matrix A:
! 219: *
! 220: * a11 a12 a13
! 221: * a22 a23 a24
! 222: * a33 a34 a35
! 223: * a44 a45 a46
! 224: * a55 a56
! 225: * (aij=conjg(aji)) a66
! 226: *
! 227: * Band storage of the upper triangle of A:
! 228: *
! 229: * * * a13 a24 a35 a46
! 230: * * a12 a23 a34 a45 a56
! 231: * a11 a22 a33 a44 a55 a66
! 232: *
! 233: * Similarly, if UPLO = 'L' the format of A is as follows:
! 234: *
! 235: * a11 a22 a33 a44 a55 a66
! 236: * a21 a32 a43 a54 a65 *
! 237: * a31 a42 a53 a64 * *
! 238: *
! 239: * Array elements marked * are not used by the routine.
! 240: *
! 241: * =====================================================================
! 242: *
! 243: * .. Parameters ..
! 244: DOUBLE PRECISION ZERO, ONE
! 245: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
! 246: * ..
! 247: * .. Local Scalars ..
! 248: LOGICAL EQUIL, NOFACT, RCEQU, UPPER
! 249: INTEGER I, INFEQU, J, J1, J2
! 250: DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
! 251: * ..
! 252: * .. External Functions ..
! 253: LOGICAL LSAME
! 254: DOUBLE PRECISION DLAMCH, DLANSB
! 255: EXTERNAL LSAME, DLAMCH, DLANSB
! 256: * ..
! 257: * .. External Subroutines ..
! 258: EXTERNAL DCOPY, DLACPY, DLAQSB, DPBCON, DPBEQU, DPBRFS,
! 259: $ DPBTRF, DPBTRS, XERBLA
! 260: * ..
! 261: * .. Intrinsic Functions ..
! 262: INTRINSIC MAX, MIN
! 263: * ..
! 264: * .. Executable Statements ..
! 265: *
! 266: INFO = 0
! 267: NOFACT = LSAME( FACT, 'N' )
! 268: EQUIL = LSAME( FACT, 'E' )
! 269: UPPER = LSAME( UPLO, 'U' )
! 270: IF( NOFACT .OR. EQUIL ) THEN
! 271: EQUED = 'N'
! 272: RCEQU = .FALSE.
! 273: ELSE
! 274: RCEQU = LSAME( EQUED, 'Y' )
! 275: SMLNUM = DLAMCH( 'Safe minimum' )
! 276: BIGNUM = ONE / SMLNUM
! 277: END IF
! 278: *
! 279: * Test the input parameters.
! 280: *
! 281: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
! 282: $ THEN
! 283: INFO = -1
! 284: ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
! 285: INFO = -2
! 286: ELSE IF( N.LT.0 ) THEN
! 287: INFO = -3
! 288: ELSE IF( KD.LT.0 ) THEN
! 289: INFO = -4
! 290: ELSE IF( NRHS.LT.0 ) THEN
! 291: INFO = -5
! 292: ELSE IF( LDAB.LT.KD+1 ) THEN
! 293: INFO = -7
! 294: ELSE IF( LDAFB.LT.KD+1 ) THEN
! 295: INFO = -9
! 296: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
! 297: $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
! 298: INFO = -10
! 299: ELSE
! 300: IF( RCEQU ) THEN
! 301: SMIN = BIGNUM
! 302: SMAX = ZERO
! 303: DO 10 J = 1, N
! 304: SMIN = MIN( SMIN, S( J ) )
! 305: SMAX = MAX( SMAX, S( J ) )
! 306: 10 CONTINUE
! 307: IF( SMIN.LE.ZERO ) THEN
! 308: INFO = -11
! 309: ELSE IF( N.GT.0 ) THEN
! 310: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
! 311: ELSE
! 312: SCOND = ONE
! 313: END IF
! 314: END IF
! 315: IF( INFO.EQ.0 ) THEN
! 316: IF( LDB.LT.MAX( 1, N ) ) THEN
! 317: INFO = -13
! 318: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
! 319: INFO = -15
! 320: END IF
! 321: END IF
! 322: END IF
! 323: *
! 324: IF( INFO.NE.0 ) THEN
! 325: CALL XERBLA( 'DPBSVX', -INFO )
! 326: RETURN
! 327: END IF
! 328: *
! 329: IF( EQUIL ) THEN
! 330: *
! 331: * Compute row and column scalings to equilibrate the matrix A.
! 332: *
! 333: CALL DPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFEQU )
! 334: IF( INFEQU.EQ.0 ) THEN
! 335: *
! 336: * Equilibrate the matrix.
! 337: *
! 338: CALL DLAQSB( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED )
! 339: RCEQU = LSAME( EQUED, 'Y' )
! 340: END IF
! 341: END IF
! 342: *
! 343: * Scale the right-hand side.
! 344: *
! 345: IF( RCEQU ) THEN
! 346: DO 30 J = 1, NRHS
! 347: DO 20 I = 1, N
! 348: B( I, J ) = S( I )*B( I, J )
! 349: 20 CONTINUE
! 350: 30 CONTINUE
! 351: END IF
! 352: *
! 353: IF( NOFACT .OR. EQUIL ) THEN
! 354: *
! 355: * Compute the Cholesky factorization A = U'*U or A = L*L'.
! 356: *
! 357: IF( UPPER ) THEN
! 358: DO 40 J = 1, N
! 359: J1 = MAX( J-KD, 1 )
! 360: CALL DCOPY( J-J1+1, AB( KD+1-J+J1, J ), 1,
! 361: $ AFB( KD+1-J+J1, J ), 1 )
! 362: 40 CONTINUE
! 363: ELSE
! 364: DO 50 J = 1, N
! 365: J2 = MIN( J+KD, N )
! 366: CALL DCOPY( J2-J+1, AB( 1, J ), 1, AFB( 1, J ), 1 )
! 367: 50 CONTINUE
! 368: END IF
! 369: *
! 370: CALL DPBTRF( UPLO, N, KD, AFB, LDAFB, INFO )
! 371: *
! 372: * Return if INFO is non-zero.
! 373: *
! 374: IF( INFO.GT.0 )THEN
! 375: RCOND = ZERO
! 376: RETURN
! 377: END IF
! 378: END IF
! 379: *
! 380: * Compute the norm of the matrix A.
! 381: *
! 382: ANORM = DLANSB( '1', UPLO, N, KD, AB, LDAB, WORK )
! 383: *
! 384: * Compute the reciprocal of the condition number of A.
! 385: *
! 386: CALL DPBCON( UPLO, N, KD, AFB, LDAFB, ANORM, RCOND, WORK, IWORK,
! 387: $ INFO )
! 388: *
! 389: * Compute the solution matrix X.
! 390: *
! 391: CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
! 392: CALL DPBTRS( UPLO, N, KD, NRHS, AFB, LDAFB, X, LDX, INFO )
! 393: *
! 394: * Use iterative refinement to improve the computed solution and
! 395: * compute error bounds and backward error estimates for it.
! 396: *
! 397: CALL DPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X,
! 398: $ LDX, FERR, BERR, WORK, IWORK, INFO )
! 399: *
! 400: * Transform the solution matrix X to a solution of the original
! 401: * system.
! 402: *
! 403: IF( RCEQU ) THEN
! 404: DO 70 J = 1, NRHS
! 405: DO 60 I = 1, N
! 406: X( I, J ) = S( I )*X( I, J )
! 407: 60 CONTINUE
! 408: 70 CONTINUE
! 409: DO 80 J = 1, NRHS
! 410: FERR( J ) = FERR( J ) / SCOND
! 411: 80 CONTINUE
! 412: END IF
! 413: *
! 414: * Set INFO = N+1 if the matrix is singular to working precision.
! 415: *
! 416: IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
! 417: $ INFO = N + 1
! 418: *
! 419: RETURN
! 420: *
! 421: * End of DPBSVX
! 422: *
! 423: END
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