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