Annotation of rpl/lapack/lapack/dgbrfs.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
! 2: $ IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK,
! 3: $ INFO )
! 4: *
! 5: * -- LAPACK 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: * Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH.
! 11: *
! 12: * .. Scalar Arguments ..
! 13: CHARACTER TRANS
! 14: INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
! 15: * ..
! 16: * .. Array Arguments ..
! 17: INTEGER IPIV( * ), IWORK( * )
! 18: DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
! 19: $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
! 20: * ..
! 21: *
! 22: * Purpose
! 23: * =======
! 24: *
! 25: * DGBRFS improves the computed solution to a system of linear
! 26: * equations when the coefficient matrix is banded, and provides
! 27: * error bounds and backward error estimates for the solution.
! 28: *
! 29: * Arguments
! 30: * =========
! 31: *
! 32: * TRANS (input) CHARACTER*1
! 33: * Specifies the form of the system of equations:
! 34: * = 'N': A * X = B (No transpose)
! 35: * = 'T': A**T * X = B (Transpose)
! 36: * = 'C': A**H * X = B (Conjugate transpose = Transpose)
! 37: *
! 38: * N (input) INTEGER
! 39: * The order of the matrix A. N >= 0.
! 40: *
! 41: * KL (input) INTEGER
! 42: * The number of subdiagonals within the band of A. KL >= 0.
! 43: *
! 44: * KU (input) INTEGER
! 45: * The number of superdiagonals within the band of A. KU >= 0.
! 46: *
! 47: * NRHS (input) INTEGER
! 48: * The number of right hand sides, i.e., the number of columns
! 49: * of the matrices B and X. NRHS >= 0.
! 50: *
! 51: * AB (input) DOUBLE PRECISION array, dimension (LDAB,N)
! 52: * The original band matrix A, stored in rows 1 to KL+KU+1.
! 53: * The j-th column of A is stored in the j-th column of the
! 54: * array AB as follows:
! 55: * AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
! 56: *
! 57: * LDAB (input) INTEGER
! 58: * The leading dimension of the array AB. LDAB >= KL+KU+1.
! 59: *
! 60: * AFB (input) DOUBLE PRECISION array, dimension (LDAFB,N)
! 61: * Details of the LU factorization of the band matrix A, as
! 62: * computed by DGBTRF. U is stored as an upper triangular band
! 63: * matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
! 64: * the multipliers used during the factorization are stored in
! 65: * rows KL+KU+2 to 2*KL+KU+1.
! 66: *
! 67: * LDAFB (input) INTEGER
! 68: * The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1.
! 69: *
! 70: * IPIV (input) INTEGER array, dimension (N)
! 71: * The pivot indices from DGBTRF; for 1<=i<=N, row i of the
! 72: * matrix was interchanged with row IPIV(i).
! 73: *
! 74: * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
! 75: * The right hand side matrix B.
! 76: *
! 77: * LDB (input) INTEGER
! 78: * The leading dimension of the array B. LDB >= max(1,N).
! 79: *
! 80: * X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
! 81: * On entry, the solution matrix X, as computed by DGBTRS.
! 82: * On exit, the improved solution matrix X.
! 83: *
! 84: * LDX (input) INTEGER
! 85: * The leading dimension of the array X. LDX >= max(1,N).
! 86: *
! 87: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 88: * The estimated forward error bound for each solution vector
! 89: * X(j) (the j-th column of the solution matrix X).
! 90: * If XTRUE is the true solution corresponding to X(j), FERR(j)
! 91: * is an estimated upper bound for the magnitude of the largest
! 92: * element in (X(j) - XTRUE) divided by the magnitude of the
! 93: * largest element in X(j). The estimate is as reliable as
! 94: * the estimate for RCOND, and is almost always a slight
! 95: * overestimate of the true error.
! 96: *
! 97: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 98: * The componentwise relative backward error of each solution
! 99: * vector X(j) (i.e., the smallest relative change in
! 100: * any element of A or B that makes X(j) an exact solution).
! 101: *
! 102: * WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
! 103: *
! 104: * IWORK (workspace) INTEGER array, dimension (N)
! 105: *
! 106: * INFO (output) INTEGER
! 107: * = 0: successful exit
! 108: * < 0: if INFO = -i, the i-th argument had an illegal value
! 109: *
! 110: * Internal Parameters
! 111: * ===================
! 112: *
! 113: * ITMAX is the maximum number of steps of iterative refinement.
! 114: *
! 115: * =====================================================================
! 116: *
! 117: * .. Parameters ..
! 118: INTEGER ITMAX
! 119: PARAMETER ( ITMAX = 5 )
! 120: DOUBLE PRECISION ZERO
! 121: PARAMETER ( ZERO = 0.0D+0 )
! 122: DOUBLE PRECISION ONE
! 123: PARAMETER ( ONE = 1.0D+0 )
! 124: DOUBLE PRECISION TWO
! 125: PARAMETER ( TWO = 2.0D+0 )
! 126: DOUBLE PRECISION THREE
! 127: PARAMETER ( THREE = 3.0D+0 )
! 128: * ..
! 129: * .. Local Scalars ..
! 130: LOGICAL NOTRAN
! 131: CHARACTER TRANST
! 132: INTEGER COUNT, I, J, K, KASE, KK, NZ
! 133: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
! 134: * ..
! 135: * .. Local Arrays ..
! 136: INTEGER ISAVE( 3 )
! 137: * ..
! 138: * .. External Subroutines ..
! 139: EXTERNAL DAXPY, DCOPY, DGBMV, DGBTRS, DLACN2, XERBLA
! 140: * ..
! 141: * .. Intrinsic Functions ..
! 142: INTRINSIC ABS, MAX, MIN
! 143: * ..
! 144: * .. External Functions ..
! 145: LOGICAL LSAME
! 146: DOUBLE PRECISION DLAMCH
! 147: EXTERNAL LSAME, DLAMCH
! 148: * ..
! 149: * .. Executable Statements ..
! 150: *
! 151: * Test the input parameters.
! 152: *
! 153: INFO = 0
! 154: NOTRAN = LSAME( TRANS, 'N' )
! 155: IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
! 156: $ LSAME( TRANS, 'C' ) ) THEN
! 157: INFO = -1
! 158: ELSE IF( N.LT.0 ) THEN
! 159: INFO = -2
! 160: ELSE IF( KL.LT.0 ) THEN
! 161: INFO = -3
! 162: ELSE IF( KU.LT.0 ) THEN
! 163: INFO = -4
! 164: ELSE IF( NRHS.LT.0 ) THEN
! 165: INFO = -5
! 166: ELSE IF( LDAB.LT.KL+KU+1 ) THEN
! 167: INFO = -7
! 168: ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
! 169: INFO = -9
! 170: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
! 171: INFO = -12
! 172: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
! 173: INFO = -14
! 174: END IF
! 175: IF( INFO.NE.0 ) THEN
! 176: CALL XERBLA( 'DGBRFS', -INFO )
! 177: RETURN
! 178: END IF
! 179: *
! 180: * Quick return if possible
! 181: *
! 182: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
! 183: DO 10 J = 1, NRHS
! 184: FERR( J ) = ZERO
! 185: BERR( J ) = ZERO
! 186: 10 CONTINUE
! 187: RETURN
! 188: END IF
! 189: *
! 190: IF( NOTRAN ) THEN
! 191: TRANST = 'T'
! 192: ELSE
! 193: TRANST = 'N'
! 194: END IF
! 195: *
! 196: * NZ = maximum number of nonzero elements in each row of A, plus 1
! 197: *
! 198: NZ = MIN( KL+KU+2, N+1 )
! 199: EPS = DLAMCH( 'Epsilon' )
! 200: SAFMIN = DLAMCH( 'Safe minimum' )
! 201: SAFE1 = NZ*SAFMIN
! 202: SAFE2 = SAFE1 / EPS
! 203: *
! 204: * Do for each right hand side
! 205: *
! 206: DO 140 J = 1, NRHS
! 207: *
! 208: COUNT = 1
! 209: LSTRES = THREE
! 210: 20 CONTINUE
! 211: *
! 212: * Loop until stopping criterion is satisfied.
! 213: *
! 214: * Compute residual R = B - op(A) * X,
! 215: * where op(A) = A, A**T, or A**H, depending on TRANS.
! 216: *
! 217: CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
! 218: CALL DGBMV( TRANS, N, N, KL, KU, -ONE, AB, LDAB, X( 1, J ), 1,
! 219: $ ONE, WORK( N+1 ), 1 )
! 220: *
! 221: * Compute componentwise relative backward error from formula
! 222: *
! 223: * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
! 224: *
! 225: * where abs(Z) is the componentwise absolute value of the matrix
! 226: * or vector Z. If the i-th component of the denominator is less
! 227: * than SAFE2, then SAFE1 is added to the i-th components of the
! 228: * numerator and denominator before dividing.
! 229: *
! 230: DO 30 I = 1, N
! 231: WORK( I ) = ABS( B( I, J ) )
! 232: 30 CONTINUE
! 233: *
! 234: * Compute abs(op(A))*abs(X) + abs(B).
! 235: *
! 236: IF( NOTRAN ) THEN
! 237: DO 50 K = 1, N
! 238: KK = KU + 1 - K
! 239: XK = ABS( X( K, J ) )
! 240: DO 40 I = MAX( 1, K-KU ), MIN( N, K+KL )
! 241: WORK( I ) = WORK( I ) + ABS( AB( KK+I, K ) )*XK
! 242: 40 CONTINUE
! 243: 50 CONTINUE
! 244: ELSE
! 245: DO 70 K = 1, N
! 246: S = ZERO
! 247: KK = KU + 1 - K
! 248: DO 60 I = MAX( 1, K-KU ), MIN( N, K+KL )
! 249: S = S + ABS( AB( KK+I, K ) )*ABS( X( I, J ) )
! 250: 60 CONTINUE
! 251: WORK( K ) = WORK( K ) + S
! 252: 70 CONTINUE
! 253: END IF
! 254: S = ZERO
! 255: DO 80 I = 1, N
! 256: IF( WORK( I ).GT.SAFE2 ) THEN
! 257: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
! 258: ELSE
! 259: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
! 260: $ ( WORK( I )+SAFE1 ) )
! 261: END IF
! 262: 80 CONTINUE
! 263: BERR( J ) = S
! 264: *
! 265: * Test stopping criterion. Continue iterating if
! 266: * 1) The residual BERR(J) is larger than machine epsilon, and
! 267: * 2) BERR(J) decreased by at least a factor of 2 during the
! 268: * last iteration, and
! 269: * 3) At most ITMAX iterations tried.
! 270: *
! 271: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
! 272: $ COUNT.LE.ITMAX ) THEN
! 273: *
! 274: * Update solution and try again.
! 275: *
! 276: CALL DGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV,
! 277: $ WORK( N+1 ), N, INFO )
! 278: CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
! 279: LSTRES = BERR( J )
! 280: COUNT = COUNT + 1
! 281: GO TO 20
! 282: END IF
! 283: *
! 284: * Bound error from formula
! 285: *
! 286: * norm(X - XTRUE) / norm(X) .le. FERR =
! 287: * norm( abs(inv(op(A)))*
! 288: * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
! 289: *
! 290: * where
! 291: * norm(Z) is the magnitude of the largest component of Z
! 292: * inv(op(A)) is the inverse of op(A)
! 293: * abs(Z) is the componentwise absolute value of the matrix or
! 294: * vector Z
! 295: * NZ is the maximum number of nonzeros in any row of A, plus 1
! 296: * EPS is machine epsilon
! 297: *
! 298: * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
! 299: * is incremented by SAFE1 if the i-th component of
! 300: * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
! 301: *
! 302: * Use DLACN2 to estimate the infinity-norm of the matrix
! 303: * inv(op(A)) * diag(W),
! 304: * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
! 305: *
! 306: DO 90 I = 1, N
! 307: IF( WORK( I ).GT.SAFE2 ) THEN
! 308: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
! 309: ELSE
! 310: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
! 311: END IF
! 312: 90 CONTINUE
! 313: *
! 314: KASE = 0
! 315: 100 CONTINUE
! 316: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
! 317: $ KASE, ISAVE )
! 318: IF( KASE.NE.0 ) THEN
! 319: IF( KASE.EQ.1 ) THEN
! 320: *
! 321: * Multiply by diag(W)*inv(op(A)**T).
! 322: *
! 323: CALL DGBTRS( TRANST, N, KL, KU, 1, AFB, LDAFB, IPIV,
! 324: $ WORK( N+1 ), N, INFO )
! 325: DO 110 I = 1, N
! 326: WORK( N+I ) = WORK( N+I )*WORK( I )
! 327: 110 CONTINUE
! 328: ELSE
! 329: *
! 330: * Multiply by inv(op(A))*diag(W).
! 331: *
! 332: DO 120 I = 1, N
! 333: WORK( N+I ) = WORK( N+I )*WORK( I )
! 334: 120 CONTINUE
! 335: CALL DGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV,
! 336: $ WORK( N+1 ), N, INFO )
! 337: END IF
! 338: GO TO 100
! 339: END IF
! 340: *
! 341: * Normalize error.
! 342: *
! 343: LSTRES = ZERO
! 344: DO 130 I = 1, N
! 345: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
! 346: 130 CONTINUE
! 347: IF( LSTRES.NE.ZERO )
! 348: $ FERR( J ) = FERR( J ) / LSTRES
! 349: *
! 350: 140 CONTINUE
! 351: *
! 352: RETURN
! 353: *
! 354: * End of DGBRFS
! 355: *
! 356: END
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