00001 SUBROUTINE ZSYTF2( UPLO, N, A, LDA, IPIV, INFO ) 00002 * 00003 * -- LAPACK routine (version 3.2) -- 00004 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00005 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00006 * November 2006 00007 * 00008 * .. Scalar Arguments .. 00009 CHARACTER UPLO 00010 INTEGER INFO, LDA, N 00011 * .. 00012 * .. Array Arguments .. 00013 INTEGER IPIV( * ) 00014 COMPLEX*16 A( LDA, * ) 00015 * .. 00016 * 00017 * Purpose 00018 * ======= 00019 * 00020 * ZSYTF2 computes the factorization of a complex symmetric matrix A 00021 * using the Bunch-Kaufman diagonal pivoting method: 00022 * 00023 * A = U*D*U' or A = L*D*L' 00024 * 00025 * where U (or L) is a product of permutation and unit upper (lower) 00026 * triangular matrices, U' is the transpose of U, and D is symmetric and 00027 * block diagonal with 1-by-1 and 2-by-2 diagonal blocks. 00028 * 00029 * This is the unblocked version of the algorithm, calling Level 2 BLAS. 00030 * 00031 * Arguments 00032 * ========= 00033 * 00034 * UPLO (input) CHARACTER*1 00035 * Specifies whether the upper or lower triangular part of the 00036 * symmetric matrix A is stored: 00037 * = 'U': Upper triangular 00038 * = 'L': Lower triangular 00039 * 00040 * N (input) INTEGER 00041 * The order of the matrix A. N >= 0. 00042 * 00043 * A (input/output) COMPLEX*16 array, dimension (LDA,N) 00044 * On entry, the symmetric matrix A. If UPLO = 'U', the leading 00045 * n-by-n upper triangular part of A contains the upper 00046 * triangular part of the matrix A, and the strictly lower 00047 * triangular part of A is not referenced. If UPLO = 'L', the 00048 * leading n-by-n lower triangular part of A contains the lower 00049 * triangular part of the matrix A, and the strictly upper 00050 * triangular part of A is not referenced. 00051 * 00052 * On exit, the block diagonal matrix D and the multipliers used 00053 * to obtain the factor U or L (see below for further details). 00054 * 00055 * LDA (input) INTEGER 00056 * The leading dimension of the array A. LDA >= max(1,N). 00057 * 00058 * IPIV (output) INTEGER array, dimension (N) 00059 * Details of the interchanges and the block structure of D. 00060 * If IPIV(k) > 0, then rows and columns k and IPIV(k) were 00061 * interchanged and D(k,k) is a 1-by-1 diagonal block. 00062 * If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and 00063 * columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) 00064 * is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = 00065 * IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were 00066 * interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. 00067 * 00068 * INFO (output) INTEGER 00069 * = 0: successful exit 00070 * < 0: if INFO = -k, the k-th argument had an illegal value 00071 * > 0: if INFO = k, D(k,k) is exactly zero. The factorization 00072 * has been completed, but the block diagonal matrix D is 00073 * exactly singular, and division by zero will occur if it 00074 * is used to solve a system of equations. 00075 * 00076 * Further Details 00077 * =============== 00078 * 00079 * 09-29-06 - patch from 00080 * Bobby Cheng, MathWorks 00081 * 00082 * Replace l.209 and l.377 00083 * IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN 00084 * by 00085 * IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN 00086 * 00087 * 1-96 - Based on modifications by J. Lewis, Boeing Computer Services 00088 * Company 00089 * 00090 * If UPLO = 'U', then A = U*D*U', where 00091 * U = P(n)*U(n)* ... *P(k)U(k)* ..., 00092 * i.e., U is a product of terms P(k)*U(k), where k decreases from n to 00093 * 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 00094 * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as 00095 * defined by IPIV(k), and U(k) is a unit upper triangular matrix, such 00096 * that if the diagonal block D(k) is of order s (s = 1 or 2), then 00097 * 00098 * ( I v 0 ) k-s 00099 * U(k) = ( 0 I 0 ) s 00100 * ( 0 0 I ) n-k 00101 * k-s s n-k 00102 * 00103 * If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). 00104 * If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), 00105 * and A(k,k), and v overwrites A(1:k-2,k-1:k). 00106 * 00107 * If UPLO = 'L', then A = L*D*L', where 00108 * L = P(1)*L(1)* ... *P(k)*L(k)* ..., 00109 * i.e., L is a product of terms P(k)*L(k), where k increases from 1 to 00110 * n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 00111 * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as 00112 * defined by IPIV(k), and L(k) is a unit lower triangular matrix, such 00113 * that if the diagonal block D(k) is of order s (s = 1 or 2), then 00114 * 00115 * ( I 0 0 ) k-1 00116 * L(k) = ( 0 I 0 ) s 00117 * ( 0 v I ) n-k-s+1 00118 * k-1 s n-k-s+1 00119 * 00120 * If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). 00121 * If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), 00122 * and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1). 00123 * 00124 * ===================================================================== 00125 * 00126 * .. Parameters .. 00127 DOUBLE PRECISION ZERO, ONE 00128 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00129 DOUBLE PRECISION EIGHT, SEVTEN 00130 PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 ) 00131 COMPLEX*16 CONE 00132 PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) ) 00133 * .. 00134 * .. Local Scalars .. 00135 LOGICAL UPPER 00136 INTEGER I, IMAX, J, JMAX, K, KK, KP, KSTEP 00137 DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, ROWMAX 00138 COMPLEX*16 D11, D12, D21, D22, R1, T, WK, WKM1, WKP1, Z 00139 * .. 00140 * .. External Functions .. 00141 LOGICAL DISNAN, LSAME 00142 INTEGER IZAMAX 00143 EXTERNAL DISNAN, LSAME, IZAMAX 00144 * .. 00145 * .. External Subroutines .. 00146 EXTERNAL XERBLA, ZSCAL, ZSWAP, ZSYR 00147 * .. 00148 * .. Intrinsic Functions .. 00149 INTRINSIC ABS, DBLE, DIMAG, MAX, SQRT 00150 * .. 00151 * .. Statement Functions .. 00152 DOUBLE PRECISION CABS1 00153 * .. 00154 * .. Statement Function definitions .. 00155 CABS1( Z ) = ABS( DBLE( Z ) ) + ABS( DIMAG( Z ) ) 00156 * .. 00157 * .. Executable Statements .. 00158 * 00159 * Test the input parameters. 00160 * 00161 INFO = 0 00162 UPPER = LSAME( UPLO, 'U' ) 00163 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 00164 INFO = -1 00165 ELSE IF( N.LT.0 ) THEN 00166 INFO = -2 00167 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00168 INFO = -4 00169 END IF 00170 IF( INFO.NE.0 ) THEN 00171 CALL XERBLA( 'ZSYTF2', -INFO ) 00172 RETURN 00173 END IF 00174 * 00175 * Initialize ALPHA for use in choosing pivot block size. 00176 * 00177 ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT 00178 * 00179 IF( UPPER ) THEN 00180 * 00181 * Factorize A as U*D*U' using the upper triangle of A 00182 * 00183 * K is the main loop index, decreasing from N to 1 in steps of 00184 * 1 or 2 00185 * 00186 K = N 00187 10 CONTINUE 00188 * 00189 * If K < 1, exit from loop 00190 * 00191 IF( K.LT.1 ) 00192 $ GO TO 70 00193 KSTEP = 1 00194 * 00195 * Determine rows and columns to be interchanged and whether 00196 * a 1-by-1 or 2-by-2 pivot block will be used 00197 * 00198 ABSAKK = CABS1( A( K, K ) ) 00199 * 00200 * IMAX is the row-index of the largest off-diagonal element in 00201 * column K, and COLMAX is its absolute value 00202 * 00203 IF( K.GT.1 ) THEN 00204 IMAX = IZAMAX( K-1, A( 1, K ), 1 ) 00205 COLMAX = CABS1( A( IMAX, K ) ) 00206 ELSE 00207 COLMAX = ZERO 00208 END IF 00209 * 00210 IF( MAX( ABSAKK, COLMAX ).EQ.ZERO .OR. DISNAN(ABSAKK) ) THEN 00211 * 00212 * Column K is zero or contains a NaN: set INFO and continue 00213 * 00214 IF( INFO.EQ.0 ) 00215 $ INFO = K 00216 KP = K 00217 ELSE 00218 IF( ABSAKK.GE.ALPHA*COLMAX ) THEN 00219 * 00220 * no interchange, use 1-by-1 pivot block 00221 * 00222 KP = K 00223 ELSE 00224 * 00225 * JMAX is the column-index of the largest off-diagonal 00226 * element in row IMAX, and ROWMAX is its absolute value 00227 * 00228 JMAX = IMAX + IZAMAX( K-IMAX, A( IMAX, IMAX+1 ), LDA ) 00229 ROWMAX = CABS1( A( IMAX, JMAX ) ) 00230 IF( IMAX.GT.1 ) THEN 00231 JMAX = IZAMAX( IMAX-1, A( 1, IMAX ), 1 ) 00232 ROWMAX = MAX( ROWMAX, CABS1( A( JMAX, IMAX ) ) ) 00233 END IF 00234 * 00235 IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN 00236 * 00237 * no interchange, use 1-by-1 pivot block 00238 * 00239 KP = K 00240 ELSE IF( CABS1( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN 00241 * 00242 * interchange rows and columns K and IMAX, use 1-by-1 00243 * pivot block 00244 * 00245 KP = IMAX 00246 ELSE 00247 * 00248 * interchange rows and columns K-1 and IMAX, use 2-by-2 00249 * pivot block 00250 * 00251 KP = IMAX 00252 KSTEP = 2 00253 END IF 00254 END IF 00255 * 00256 KK = K - KSTEP + 1 00257 IF( KP.NE.KK ) THEN 00258 * 00259 * Interchange rows and columns KK and KP in the leading 00260 * submatrix A(1:k,1:k) 00261 * 00262 CALL ZSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 ) 00263 CALL ZSWAP( KK-KP-1, A( KP+1, KK ), 1, A( KP, KP+1 ), 00264 $ LDA ) 00265 T = A( KK, KK ) 00266 A( KK, KK ) = A( KP, KP ) 00267 A( KP, KP ) = T 00268 IF( KSTEP.EQ.2 ) THEN 00269 T = A( K-1, K ) 00270 A( K-1, K ) = A( KP, K ) 00271 A( KP, K ) = T 00272 END IF 00273 END IF 00274 * 00275 * Update the leading submatrix 00276 * 00277 IF( KSTEP.EQ.1 ) THEN 00278 * 00279 * 1-by-1 pivot block D(k): column k now holds 00280 * 00281 * W(k) = U(k)*D(k) 00282 * 00283 * where U(k) is the k-th column of U 00284 * 00285 * Perform a rank-1 update of A(1:k-1,1:k-1) as 00286 * 00287 * A := A - U(k)*D(k)*U(k)' = A - W(k)*1/D(k)*W(k)' 00288 * 00289 R1 = CONE / A( K, K ) 00290 CALL ZSYR( UPLO, K-1, -R1, A( 1, K ), 1, A, LDA ) 00291 * 00292 * Store U(k) in column k 00293 * 00294 CALL ZSCAL( K-1, R1, A( 1, K ), 1 ) 00295 ELSE 00296 * 00297 * 2-by-2 pivot block D(k): columns k and k-1 now hold 00298 * 00299 * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k) 00300 * 00301 * where U(k) and U(k-1) are the k-th and (k-1)-th columns 00302 * of U 00303 * 00304 * Perform a rank-2 update of A(1:k-2,1:k-2) as 00305 * 00306 * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )' 00307 * = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )' 00308 * 00309 IF( K.GT.2 ) THEN 00310 * 00311 D12 = A( K-1, K ) 00312 D22 = A( K-1, K-1 ) / D12 00313 D11 = A( K, K ) / D12 00314 T = CONE / ( D11*D22-CONE ) 00315 D12 = T / D12 00316 * 00317 DO 30 J = K - 2, 1, -1 00318 WKM1 = D12*( D11*A( J, K-1 )-A( J, K ) ) 00319 WK = D12*( D22*A( J, K )-A( J, K-1 ) ) 00320 DO 20 I = J, 1, -1 00321 A( I, J ) = A( I, J ) - A( I, K )*WK - 00322 $ A( I, K-1 )*WKM1 00323 20 CONTINUE 00324 A( J, K ) = WK 00325 A( J, K-1 ) = WKM1 00326 30 CONTINUE 00327 * 00328 END IF 00329 * 00330 END IF 00331 END IF 00332 * 00333 * Store details of the interchanges in IPIV 00334 * 00335 IF( KSTEP.EQ.1 ) THEN 00336 IPIV( K ) = KP 00337 ELSE 00338 IPIV( K ) = -KP 00339 IPIV( K-1 ) = -KP 00340 END IF 00341 * 00342 * Decrease K and return to the start of the main loop 00343 * 00344 K = K - KSTEP 00345 GO TO 10 00346 * 00347 ELSE 00348 * 00349 * Factorize A as L*D*L' using the lower triangle of A 00350 * 00351 * K is the main loop index, increasing from 1 to N in steps of 00352 * 1 or 2 00353 * 00354 K = 1 00355 40 CONTINUE 00356 * 00357 * If K > N, exit from loop 00358 * 00359 IF( K.GT.N ) 00360 $ GO TO 70 00361 KSTEP = 1 00362 * 00363 * Determine rows and columns to be interchanged and whether 00364 * a 1-by-1 or 2-by-2 pivot block will be used 00365 * 00366 ABSAKK = CABS1( A( K, K ) ) 00367 * 00368 * IMAX is the row-index of the largest off-diagonal element in 00369 * column K, and COLMAX is its absolute value 00370 * 00371 IF( K.LT.N ) THEN 00372 IMAX = K + IZAMAX( N-K, A( K+1, K ), 1 ) 00373 COLMAX = CABS1( A( IMAX, K ) ) 00374 ELSE 00375 COLMAX = ZERO 00376 END IF 00377 * 00378 IF( MAX( ABSAKK, COLMAX ).EQ.ZERO .OR. DISNAN(ABSAKK) ) THEN 00379 * 00380 * Column K is zero or contains a NaN: set INFO and continue 00381 * 00382 IF( INFO.EQ.0 ) 00383 $ INFO = K 00384 KP = K 00385 ELSE 00386 IF( ABSAKK.GE.ALPHA*COLMAX ) THEN 00387 * 00388 * no interchange, use 1-by-1 pivot block 00389 * 00390 KP = K 00391 ELSE 00392 * 00393 * JMAX is the column-index of the largest off-diagonal 00394 * element in row IMAX, and ROWMAX is its absolute value 00395 * 00396 JMAX = K - 1 + IZAMAX( IMAX-K, A( IMAX, K ), LDA ) 00397 ROWMAX = CABS1( A( IMAX, JMAX ) ) 00398 IF( IMAX.LT.N ) THEN 00399 JMAX = IMAX + IZAMAX( N-IMAX, A( IMAX+1, IMAX ), 1 ) 00400 ROWMAX = MAX( ROWMAX, CABS1( A( JMAX, IMAX ) ) ) 00401 END IF 00402 * 00403 IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN 00404 * 00405 * no interchange, use 1-by-1 pivot block 00406 * 00407 KP = K 00408 ELSE IF( CABS1( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN 00409 * 00410 * interchange rows and columns K and IMAX, use 1-by-1 00411 * pivot block 00412 * 00413 KP = IMAX 00414 ELSE 00415 * 00416 * interchange rows and columns K+1 and IMAX, use 2-by-2 00417 * pivot block 00418 * 00419 KP = IMAX 00420 KSTEP = 2 00421 END IF 00422 END IF 00423 * 00424 KK = K + KSTEP - 1 00425 IF( KP.NE.KK ) THEN 00426 * 00427 * Interchange rows and columns KK and KP in the trailing 00428 * submatrix A(k:n,k:n) 00429 * 00430 IF( KP.LT.N ) 00431 $ CALL ZSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 ) 00432 CALL ZSWAP( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ), 00433 $ LDA ) 00434 T = A( KK, KK ) 00435 A( KK, KK ) = A( KP, KP ) 00436 A( KP, KP ) = T 00437 IF( KSTEP.EQ.2 ) THEN 00438 T = A( K+1, K ) 00439 A( K+1, K ) = A( KP, K ) 00440 A( KP, K ) = T 00441 END IF 00442 END IF 00443 * 00444 * Update the trailing submatrix 00445 * 00446 IF( KSTEP.EQ.1 ) THEN 00447 * 00448 * 1-by-1 pivot block D(k): column k now holds 00449 * 00450 * W(k) = L(k)*D(k) 00451 * 00452 * where L(k) is the k-th column of L 00453 * 00454 IF( K.LT.N ) THEN 00455 * 00456 * Perform a rank-1 update of A(k+1:n,k+1:n) as 00457 * 00458 * A := A - L(k)*D(k)*L(k)' = A - W(k)*(1/D(k))*W(k)' 00459 * 00460 R1 = CONE / A( K, K ) 00461 CALL ZSYR( UPLO, N-K, -R1, A( K+1, K ), 1, 00462 $ A( K+1, K+1 ), LDA ) 00463 * 00464 * Store L(k) in column K 00465 * 00466 CALL ZSCAL( N-K, R1, A( K+1, K ), 1 ) 00467 END IF 00468 ELSE 00469 * 00470 * 2-by-2 pivot block D(k) 00471 * 00472 IF( K.LT.N-1 ) THEN 00473 * 00474 * Perform a rank-2 update of A(k+2:n,k+2:n) as 00475 * 00476 * A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )' 00477 * = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )' 00478 * 00479 * where L(k) and L(k+1) are the k-th and (k+1)-th 00480 * columns of L 00481 * 00482 D21 = A( K+1, K ) 00483 D11 = A( K+1, K+1 ) / D21 00484 D22 = A( K, K ) / D21 00485 T = CONE / ( D11*D22-CONE ) 00486 D21 = T / D21 00487 * 00488 DO 60 J = K + 2, N 00489 WK = D21*( D11*A( J, K )-A( J, K+1 ) ) 00490 WKP1 = D21*( D22*A( J, K+1 )-A( J, K ) ) 00491 DO 50 I = J, N 00492 A( I, J ) = A( I, J ) - A( I, K )*WK - 00493 $ A( I, K+1 )*WKP1 00494 50 CONTINUE 00495 A( J, K ) = WK 00496 A( J, K+1 ) = WKP1 00497 60 CONTINUE 00498 END IF 00499 END IF 00500 END IF 00501 * 00502 * Store details of the interchanges in IPIV 00503 * 00504 IF( KSTEP.EQ.1 ) THEN 00505 IPIV( K ) = KP 00506 ELSE 00507 IPIV( K ) = -KP 00508 IPIV( K+1 ) = -KP 00509 END IF 00510 * 00511 * Increase K and return to the start of the main loop 00512 * 00513 K = K + KSTEP 00514 GO TO 40 00515 * 00516 END IF 00517 * 00518 70 CONTINUE 00519 RETURN 00520 * 00521 * End of ZSYTF2 00522 * 00523 END
1.7.2