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