Annotation of rpl/lapack/lapack/ztbrfs.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZTBRFS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B,
! 2: $ LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
! 3: *
! 4: * -- LAPACK routine (version 3.2) --
! 5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 7: * November 2006
! 8: *
! 9: * Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH.
! 10: *
! 11: * .. Scalar Arguments ..
! 12: CHARACTER DIAG, TRANS, UPLO
! 13: INTEGER INFO, KD, LDAB, LDB, LDX, N, NRHS
! 14: * ..
! 15: * .. Array Arguments ..
! 16: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
! 17: COMPLEX*16 AB( LDAB, * ), B( LDB, * ), WORK( * ),
! 18: $ X( LDX, * )
! 19: * ..
! 20: *
! 21: * Purpose
! 22: * =======
! 23: *
! 24: * ZTBRFS provides error bounds and backward error estimates for the
! 25: * solution to a system of linear equations with a triangular band
! 26: * coefficient matrix.
! 27: *
! 28: * The solution matrix X must be computed by ZTBTRS or some other
! 29: * means before entering this routine. ZTBRFS does not do iterative
! 30: * refinement because doing so cannot improve the backward error.
! 31: *
! 32: * Arguments
! 33: * =========
! 34: *
! 35: * UPLO (input) CHARACTER*1
! 36: * = 'U': A is upper triangular;
! 37: * = 'L': A is lower triangular.
! 38: *
! 39: * TRANS (input) CHARACTER*1
! 40: * Specifies the form of the system of equations:
! 41: * = 'N': A * X = B (No transpose)
! 42: * = 'T': A**T * X = B (Transpose)
! 43: * = 'C': A**H * X = B (Conjugate transpose)
! 44: *
! 45: * DIAG (input) CHARACTER*1
! 46: * = 'N': A is non-unit triangular;
! 47: * = 'U': A is unit triangular.
! 48: *
! 49: * N (input) INTEGER
! 50: * The order of the matrix A. N >= 0.
! 51: *
! 52: * KD (input) INTEGER
! 53: * The number of superdiagonals or subdiagonals of the
! 54: * triangular band matrix A. KD >= 0.
! 55: *
! 56: * NRHS (input) INTEGER
! 57: * The number of right hand sides, i.e., the number of columns
! 58: * of the matrices B and X. NRHS >= 0.
! 59: *
! 60: * AB (input) COMPLEX*16 array, dimension (LDAB,N)
! 61: * The upper or lower triangular band matrix A, stored in the
! 62: * first kd+1 rows of the array. The j-th column of A is stored
! 63: * in the j-th column of the array AB as follows:
! 64: * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
! 65: * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
! 66: * If DIAG = 'U', the diagonal elements of A are not referenced
! 67: * and are assumed to be 1.
! 68: *
! 69: * LDAB (input) INTEGER
! 70: * The leading dimension of the array AB. LDAB >= KD+1.
! 71: *
! 72: * B (input) COMPLEX*16 array, dimension (LDB,NRHS)
! 73: * The right hand side matrix B.
! 74: *
! 75: * LDB (input) INTEGER
! 76: * The leading dimension of the array B. LDB >= max(1,N).
! 77: *
! 78: * X (input) COMPLEX*16 array, dimension (LDX,NRHS)
! 79: * The solution matrix X.
! 80: *
! 81: * LDX (input) INTEGER
! 82: * The leading dimension of the array X. LDX >= max(1,N).
! 83: *
! 84: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 85: * The estimated forward error bound for each solution vector
! 86: * X(j) (the j-th column of the solution matrix X).
! 87: * If XTRUE is the true solution corresponding to X(j), FERR(j)
! 88: * is an estimated upper bound for the magnitude of the largest
! 89: * element in (X(j) - XTRUE) divided by the magnitude of the
! 90: * largest element in X(j). The estimate is as reliable as
! 91: * the estimate for RCOND, and is almost always a slight
! 92: * overestimate of the true error.
! 93: *
! 94: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
! 95: * The componentwise relative backward error of each solution
! 96: * vector X(j) (i.e., the smallest relative change in
! 97: * any element of A or B that makes X(j) an exact solution).
! 98: *
! 99: * WORK (workspace) COMPLEX*16 array, dimension (2*N)
! 100: *
! 101: * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
! 102: *
! 103: * INFO (output) INTEGER
! 104: * = 0: successful exit
! 105: * < 0: if INFO = -i, the i-th argument had an illegal value
! 106: *
! 107: * =====================================================================
! 108: *
! 109: * .. Parameters ..
! 110: DOUBLE PRECISION ZERO
! 111: PARAMETER ( ZERO = 0.0D+0 )
! 112: COMPLEX*16 ONE
! 113: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
! 114: * ..
! 115: * .. Local Scalars ..
! 116: LOGICAL NOTRAN, NOUNIT, UPPER
! 117: CHARACTER TRANSN, TRANST
! 118: INTEGER I, J, K, KASE, NZ
! 119: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
! 120: COMPLEX*16 ZDUM
! 121: * ..
! 122: * .. Local Arrays ..
! 123: INTEGER ISAVE( 3 )
! 124: * ..
! 125: * .. External Subroutines ..
! 126: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZLACN2, ZTBMV, ZTBSV
! 127: * ..
! 128: * .. Intrinsic Functions ..
! 129: INTRINSIC ABS, DBLE, DIMAG, MAX, MIN
! 130: * ..
! 131: * .. External Functions ..
! 132: LOGICAL LSAME
! 133: DOUBLE PRECISION DLAMCH
! 134: EXTERNAL LSAME, DLAMCH
! 135: * ..
! 136: * .. Statement Functions ..
! 137: DOUBLE PRECISION CABS1
! 138: * ..
! 139: * .. Statement Function definitions ..
! 140: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
! 141: * ..
! 142: * .. Executable Statements ..
! 143: *
! 144: * Test the input parameters.
! 145: *
! 146: INFO = 0
! 147: UPPER = LSAME( UPLO, 'U' )
! 148: NOTRAN = LSAME( TRANS, 'N' )
! 149: NOUNIT = LSAME( DIAG, 'N' )
! 150: *
! 151: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
! 152: INFO = -1
! 153: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
! 154: $ LSAME( TRANS, 'C' ) ) THEN
! 155: INFO = -2
! 156: ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
! 157: INFO = -3
! 158: ELSE IF( N.LT.0 ) THEN
! 159: INFO = -4
! 160: ELSE IF( KD.LT.0 ) THEN
! 161: INFO = -5
! 162: ELSE IF( NRHS.LT.0 ) THEN
! 163: INFO = -6
! 164: ELSE IF( LDAB.LT.KD+1 ) THEN
! 165: INFO = -8
! 166: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
! 167: INFO = -10
! 168: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
! 169: INFO = -12
! 170: END IF
! 171: IF( INFO.NE.0 ) THEN
! 172: CALL XERBLA( 'ZTBRFS', -INFO )
! 173: RETURN
! 174: END IF
! 175: *
! 176: * Quick return if possible
! 177: *
! 178: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
! 179: DO 10 J = 1, NRHS
! 180: FERR( J ) = ZERO
! 181: BERR( J ) = ZERO
! 182: 10 CONTINUE
! 183: RETURN
! 184: END IF
! 185: *
! 186: IF( NOTRAN ) THEN
! 187: TRANSN = 'N'
! 188: TRANST = 'C'
! 189: ELSE
! 190: TRANSN = 'C'
! 191: TRANST = 'N'
! 192: END IF
! 193: *
! 194: * NZ = maximum number of nonzero elements in each row of A, plus 1
! 195: *
! 196: NZ = KD + 2
! 197: EPS = DLAMCH( 'Epsilon' )
! 198: SAFMIN = DLAMCH( 'Safe minimum' )
! 199: SAFE1 = NZ*SAFMIN
! 200: SAFE2 = SAFE1 / EPS
! 201: *
! 202: * Do for each right hand side
! 203: *
! 204: DO 250 J = 1, NRHS
! 205: *
! 206: * Compute residual R = B - op(A) * X,
! 207: * where op(A) = A, A**T, or A**H, depending on TRANS.
! 208: *
! 209: CALL ZCOPY( N, X( 1, J ), 1, WORK, 1 )
! 210: CALL ZTBMV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, WORK, 1 )
! 211: CALL ZAXPY( N, -ONE, B( 1, J ), 1, WORK, 1 )
! 212: *
! 213: * Compute componentwise relative backward error from formula
! 214: *
! 215: * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
! 216: *
! 217: * where abs(Z) is the componentwise absolute value of the matrix
! 218: * or vector Z. If the i-th component of the denominator is less
! 219: * than SAFE2, then SAFE1 is added to the i-th components of the
! 220: * numerator and denominator before dividing.
! 221: *
! 222: DO 20 I = 1, N
! 223: RWORK( I ) = CABS1( B( I, J ) )
! 224: 20 CONTINUE
! 225: *
! 226: IF( NOTRAN ) THEN
! 227: *
! 228: * Compute abs(A)*abs(X) + abs(B).
! 229: *
! 230: IF( UPPER ) THEN
! 231: IF( NOUNIT ) THEN
! 232: DO 40 K = 1, N
! 233: XK = CABS1( X( K, J ) )
! 234: DO 30 I = MAX( 1, K-KD ), K
! 235: RWORK( I ) = RWORK( I ) +
! 236: $ CABS1( AB( KD+1+I-K, K ) )*XK
! 237: 30 CONTINUE
! 238: 40 CONTINUE
! 239: ELSE
! 240: DO 60 K = 1, N
! 241: XK = CABS1( X( K, J ) )
! 242: DO 50 I = MAX( 1, K-KD ), K - 1
! 243: RWORK( I ) = RWORK( I ) +
! 244: $ CABS1( AB( KD+1+I-K, K ) )*XK
! 245: 50 CONTINUE
! 246: RWORK( K ) = RWORK( K ) + XK
! 247: 60 CONTINUE
! 248: END IF
! 249: ELSE
! 250: IF( NOUNIT ) THEN
! 251: DO 80 K = 1, N
! 252: XK = CABS1( X( K, J ) )
! 253: DO 70 I = K, MIN( N, K+KD )
! 254: RWORK( I ) = RWORK( I ) +
! 255: $ CABS1( AB( 1+I-K, K ) )*XK
! 256: 70 CONTINUE
! 257: 80 CONTINUE
! 258: ELSE
! 259: DO 100 K = 1, N
! 260: XK = CABS1( X( K, J ) )
! 261: DO 90 I = K + 1, MIN( N, K+KD )
! 262: RWORK( I ) = RWORK( I ) +
! 263: $ CABS1( AB( 1+I-K, K ) )*XK
! 264: 90 CONTINUE
! 265: RWORK( K ) = RWORK( K ) + XK
! 266: 100 CONTINUE
! 267: END IF
! 268: END IF
! 269: ELSE
! 270: *
! 271: * Compute abs(A**H)*abs(X) + abs(B).
! 272: *
! 273: IF( UPPER ) THEN
! 274: IF( NOUNIT ) THEN
! 275: DO 120 K = 1, N
! 276: S = ZERO
! 277: DO 110 I = MAX( 1, K-KD ), K
! 278: S = S + CABS1( AB( KD+1+I-K, K ) )*
! 279: $ CABS1( X( I, J ) )
! 280: 110 CONTINUE
! 281: RWORK( K ) = RWORK( K ) + S
! 282: 120 CONTINUE
! 283: ELSE
! 284: DO 140 K = 1, N
! 285: S = CABS1( X( K, J ) )
! 286: DO 130 I = MAX( 1, K-KD ), K - 1
! 287: S = S + CABS1( AB( KD+1+I-K, K ) )*
! 288: $ CABS1( X( I, J ) )
! 289: 130 CONTINUE
! 290: RWORK( K ) = RWORK( K ) + S
! 291: 140 CONTINUE
! 292: END IF
! 293: ELSE
! 294: IF( NOUNIT ) THEN
! 295: DO 160 K = 1, N
! 296: S = ZERO
! 297: DO 150 I = K, MIN( N, K+KD )
! 298: S = S + CABS1( AB( 1+I-K, K ) )*
! 299: $ CABS1( X( I, J ) )
! 300: 150 CONTINUE
! 301: RWORK( K ) = RWORK( K ) + S
! 302: 160 CONTINUE
! 303: ELSE
! 304: DO 180 K = 1, N
! 305: S = CABS1( X( K, J ) )
! 306: DO 170 I = K + 1, MIN( N, K+KD )
! 307: S = S + CABS1( AB( 1+I-K, K ) )*
! 308: $ CABS1( X( I, J ) )
! 309: 170 CONTINUE
! 310: RWORK( K ) = RWORK( K ) + S
! 311: 180 CONTINUE
! 312: END IF
! 313: END IF
! 314: END IF
! 315: S = ZERO
! 316: DO 190 I = 1, N
! 317: IF( RWORK( I ).GT.SAFE2 ) THEN
! 318: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
! 319: ELSE
! 320: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
! 321: $ ( RWORK( I )+SAFE1 ) )
! 322: END IF
! 323: 190 CONTINUE
! 324: BERR( J ) = S
! 325: *
! 326: * Bound error from formula
! 327: *
! 328: * norm(X - XTRUE) / norm(X) .le. FERR =
! 329: * norm( abs(inv(op(A)))*
! 330: * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
! 331: *
! 332: * where
! 333: * norm(Z) is the magnitude of the largest component of Z
! 334: * inv(op(A)) is the inverse of op(A)
! 335: * abs(Z) is the componentwise absolute value of the matrix or
! 336: * vector Z
! 337: * NZ is the maximum number of nonzeros in any row of A, plus 1
! 338: * EPS is machine epsilon
! 339: *
! 340: * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
! 341: * is incremented by SAFE1 if the i-th component of
! 342: * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
! 343: *
! 344: * Use ZLACN2 to estimate the infinity-norm of the matrix
! 345: * inv(op(A)) * diag(W),
! 346: * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
! 347: *
! 348: DO 200 I = 1, N
! 349: IF( RWORK( I ).GT.SAFE2 ) THEN
! 350: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
! 351: ELSE
! 352: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
! 353: $ SAFE1
! 354: END IF
! 355: 200 CONTINUE
! 356: *
! 357: KASE = 0
! 358: 210 CONTINUE
! 359: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
! 360: IF( KASE.NE.0 ) THEN
! 361: IF( KASE.EQ.1 ) THEN
! 362: *
! 363: * Multiply by diag(W)*inv(op(A)**H).
! 364: *
! 365: CALL ZTBSV( UPLO, TRANST, DIAG, N, KD, AB, LDAB, WORK,
! 366: $ 1 )
! 367: DO 220 I = 1, N
! 368: WORK( I ) = RWORK( I )*WORK( I )
! 369: 220 CONTINUE
! 370: ELSE
! 371: *
! 372: * Multiply by inv(op(A))*diag(W).
! 373: *
! 374: DO 230 I = 1, N
! 375: WORK( I ) = RWORK( I )*WORK( I )
! 376: 230 CONTINUE
! 377: CALL ZTBSV( UPLO, TRANSN, DIAG, N, KD, AB, LDAB, WORK,
! 378: $ 1 )
! 379: END IF
! 380: GO TO 210
! 381: END IF
! 382: *
! 383: * Normalize error.
! 384: *
! 385: LSTRES = ZERO
! 386: DO 240 I = 1, N
! 387: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
! 388: 240 CONTINUE
! 389: IF( LSTRES.NE.ZERO )
! 390: $ FERR( J ) = FERR( J ) / LSTRES
! 391: *
! 392: 250 CONTINUE
! 393: *
! 394: RETURN
! 395: *
! 396: * End of ZTBRFS
! 397: *
! 398: END
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