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LAPACK 3.3.1
Linear Algebra PACKage
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LAPACK 3.3.1
Linear Algebra PACKage
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00001 SUBROUTINE ZHETF2( UPLO, N, A, LDA, IPIV, INFO ) 00002 * 00003 * -- LAPACK routine (version 3.3.1) -- 00004 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00005 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00006 * -- April 2011 -- 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 * ZHETF2 computes the factorization of a complex Hermitian matrix A 00021 * using the Bunch-Kaufman diagonal pivoting method: 00022 * 00023 * A = U*D*U**H or A = L*D*L**H 00024 * 00025 * where U (or L) is a product of permutation and unit upper (lower) 00026 * triangular matrices, U**H is the conjugate transpose of U, and D is 00027 * Hermitian and 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 * Hermitian 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 Hermitian 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.210 and l.393 00083 * IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN 00084 * by 00085 * IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN 00086 * 00087 * 01-01-96 - Based on modifications by 00088 * J. Lewis, Boeing Computer Services Company 00089 * A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA 00090 * 00091 * If UPLO = 'U', then A = U*D*U**H, where 00092 * U = P(n)*U(n)* ... *P(k)U(k)* ..., 00093 * i.e., U is a product of terms P(k)*U(k), where k decreases from n to 00094 * 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 00095 * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as 00096 * defined by IPIV(k), and U(k) is a unit upper triangular matrix, such 00097 * that if the diagonal block D(k) is of order s (s = 1 or 2), then 00098 * 00099 * ( I v 0 ) k-s 00100 * U(k) = ( 0 I 0 ) s 00101 * ( 0 0 I ) n-k 00102 * k-s s n-k 00103 * 00104 * If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). 00105 * If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), 00106 * and A(k,k), and v overwrites A(1:k-2,k-1:k). 00107 * 00108 * If UPLO = 'L', then A = L*D*L**H, where 00109 * L = P(1)*L(1)* ... *P(k)*L(k)* ..., 00110 * i.e., L is a product of terms P(k)*L(k), where k increases from 1 to 00111 * n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 00112 * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as 00113 * defined by IPIV(k), and L(k) is a unit lower triangular matrix, such 00114 * that if the diagonal block D(k) is of order s (s = 1 or 2), then 00115 * 00116 * ( I 0 0 ) k-1 00117 * L(k) = ( 0 I 0 ) s 00118 * ( 0 v I ) n-k-s+1 00119 * k-1 s n-k-s+1 00120 * 00121 * If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). 00122 * If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), 00123 * and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1). 00124 * 00125 * ===================================================================== 00126 * 00127 * .. Parameters .. 00128 DOUBLE PRECISION ZERO, ONE 00129 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00130 DOUBLE PRECISION EIGHT, SEVTEN 00131 PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 ) 00132 * .. 00133 * .. Local Scalars .. 00134 LOGICAL UPPER 00135 INTEGER I, IMAX, J, JMAX, K, KK, KP, KSTEP 00136 DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D, D11, D22, R1, ROWMAX, 00137 $ TT 00138 COMPLEX*16 D12, D21, T, WK, WKM1, WKP1, ZDUM 00139 * .. 00140 * .. External Functions .. 00141 LOGICAL LSAME, DISNAN 00142 INTEGER IZAMAX 00143 DOUBLE PRECISION DLAPY2 00144 EXTERNAL LSAME, IZAMAX, DLAPY2, DISNAN 00145 * .. 00146 * .. External Subroutines .. 00147 EXTERNAL XERBLA, ZDSCAL, ZHER, ZSWAP 00148 * .. 00149 * .. Intrinsic Functions .. 00150 INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX, SQRT 00151 * .. 00152 * .. Statement Functions .. 00153 DOUBLE PRECISION CABS1 00154 * .. 00155 * .. Statement Function definitions .. 00156 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) ) 00157 * .. 00158 * .. Executable Statements .. 00159 * 00160 * Test the input parameters. 00161 * 00162 INFO = 0 00163 UPPER = LSAME( UPLO, 'U' ) 00164 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 00165 INFO = -1 00166 ELSE IF( N.LT.0 ) THEN 00167 INFO = -2 00168 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00169 INFO = -4 00170 END IF 00171 IF( INFO.NE.0 ) THEN 00172 CALL XERBLA( 'ZHETF2', -INFO ) 00173 RETURN 00174 END IF 00175 * 00176 * Initialize ALPHA for use in choosing pivot block size. 00177 * 00178 ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT 00179 * 00180 IF( UPPER ) THEN 00181 * 00182 * Factorize A as U*D*U**H using the upper triangle of A 00183 * 00184 * K is the main loop index, decreasing from N to 1 in steps of 00185 * 1 or 2 00186 * 00187 K = N 00188 10 CONTINUE 00189 * 00190 * If K < 1, exit from loop 00191 * 00192 IF( K.LT.1 ) 00193 $ GO TO 90 00194 KSTEP = 1 00195 * 00196 * Determine rows and columns to be interchanged and whether 00197 * a 1-by-1 or 2-by-2 pivot block will be used 00198 * 00199 ABSAKK = ABS( DBLE( A( K, K ) ) ) 00200 * 00201 * IMAX is the row-index of the largest off-diagonal element in 00202 * column K, and COLMAX is its absolute value 00203 * 00204 IF( K.GT.1 ) THEN 00205 IMAX = IZAMAX( K-1, A( 1, K ), 1 ) 00206 COLMAX = CABS1( A( IMAX, K ) ) 00207 ELSE 00208 COLMAX = ZERO 00209 END IF 00210 * 00211 IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN 00212 * 00213 * Column K is zero or contains a NaN: set INFO and continue 00214 * 00215 IF( INFO.EQ.0 ) 00216 $ INFO = K 00217 KP = K 00218 A( K, K ) = DBLE( A( K, K ) ) 00219 ELSE 00220 IF( ABSAKK.GE.ALPHA*COLMAX ) THEN 00221 * 00222 * no interchange, use 1-by-1 pivot block 00223 * 00224 KP = K 00225 ELSE 00226 * 00227 * JMAX is the column-index of the largest off-diagonal 00228 * element in row IMAX, and ROWMAX is its absolute value 00229 * 00230 JMAX = IMAX + IZAMAX( K-IMAX, A( IMAX, IMAX+1 ), LDA ) 00231 ROWMAX = CABS1( A( IMAX, JMAX ) ) 00232 IF( IMAX.GT.1 ) THEN 00233 JMAX = IZAMAX( IMAX-1, A( 1, IMAX ), 1 ) 00234 ROWMAX = MAX( ROWMAX, CABS1( A( JMAX, IMAX ) ) ) 00235 END IF 00236 * 00237 IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN 00238 * 00239 * no interchange, use 1-by-1 pivot block 00240 * 00241 KP = K 00242 ELSE IF( ABS( DBLE( A( IMAX, IMAX ) ) ).GE.ALPHA*ROWMAX ) 00243 $ THEN 00244 * 00245 * interchange rows and columns K and IMAX, use 1-by-1 00246 * pivot block 00247 * 00248 KP = IMAX 00249 ELSE 00250 * 00251 * interchange rows and columns K-1 and IMAX, use 2-by-2 00252 * pivot block 00253 * 00254 KP = IMAX 00255 KSTEP = 2 00256 END IF 00257 END IF 00258 * 00259 KK = K - KSTEP + 1 00260 IF( KP.NE.KK ) THEN 00261 * 00262 * Interchange rows and columns KK and KP in the leading 00263 * submatrix A(1:k,1:k) 00264 * 00265 CALL ZSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 ) 00266 DO 20 J = KP + 1, KK - 1 00267 T = DCONJG( A( J, KK ) ) 00268 A( J, KK ) = DCONJG( A( KP, J ) ) 00269 A( KP, J ) = T 00270 20 CONTINUE 00271 A( KP, KK ) = DCONJG( A( KP, KK ) ) 00272 R1 = DBLE( A( KK, KK ) ) 00273 A( KK, KK ) = DBLE( A( KP, KP ) ) 00274 A( KP, KP ) = R1 00275 IF( KSTEP.EQ.2 ) THEN 00276 A( K, K ) = DBLE( A( K, K ) ) 00277 T = A( K-1, K ) 00278 A( K-1, K ) = A( KP, K ) 00279 A( KP, K ) = T 00280 END IF 00281 ELSE 00282 A( K, K ) = DBLE( A( K, K ) ) 00283 IF( KSTEP.EQ.2 ) 00284 $ A( K-1, K-1 ) = DBLE( A( K-1, K-1 ) ) 00285 END IF 00286 * 00287 * Update the leading submatrix 00288 * 00289 IF( KSTEP.EQ.1 ) THEN 00290 * 00291 * 1-by-1 pivot block D(k): column k now holds 00292 * 00293 * W(k) = U(k)*D(k) 00294 * 00295 * where U(k) is the k-th column of U 00296 * 00297 * Perform a rank-1 update of A(1:k-1,1:k-1) as 00298 * 00299 * A := A - U(k)*D(k)*U(k)**H = A - W(k)*1/D(k)*W(k)**H 00300 * 00301 R1 = ONE / DBLE( A( K, K ) ) 00302 CALL ZHER( UPLO, K-1, -R1, A( 1, K ), 1, A, LDA ) 00303 * 00304 * Store U(k) in column k 00305 * 00306 CALL ZDSCAL( K-1, R1, A( 1, K ), 1 ) 00307 ELSE 00308 * 00309 * 2-by-2 pivot block D(k): columns k and k-1 now hold 00310 * 00311 * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k) 00312 * 00313 * where U(k) and U(k-1) are the k-th and (k-1)-th columns 00314 * of U 00315 * 00316 * Perform a rank-2 update of A(1:k-2,1:k-2) as 00317 * 00318 * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**H 00319 * = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**H 00320 * 00321 IF( K.GT.2 ) THEN 00322 * 00323 D = DLAPY2( DBLE( A( K-1, K ) ), 00324 $ DIMAG( A( K-1, K ) ) ) 00325 D22 = DBLE( A( K-1, K-1 ) ) / D 00326 D11 = DBLE( A( K, K ) ) / D 00327 TT = ONE / ( D11*D22-ONE ) 00328 D12 = A( K-1, K ) / D 00329 D = TT / D 00330 * 00331 DO 40 J = K - 2, 1, -1 00332 WKM1 = D*( D11*A( J, K-1 )-DCONJG( D12 )* 00333 $ A( J, K ) ) 00334 WK = D*( D22*A( J, K )-D12*A( J, K-1 ) ) 00335 DO 30 I = J, 1, -1 00336 A( I, J ) = A( I, J ) - A( I, K )*DCONJG( WK ) - 00337 $ A( I, K-1 )*DCONJG( WKM1 ) 00338 30 CONTINUE 00339 A( J, K ) = WK 00340 A( J, K-1 ) = WKM1 00341 A( J, J ) = DCMPLX( DBLE( A( J, J ) ), 0.0D+0 ) 00342 40 CONTINUE 00343 * 00344 END IF 00345 * 00346 END IF 00347 END IF 00348 * 00349 * Store details of the interchanges in IPIV 00350 * 00351 IF( KSTEP.EQ.1 ) THEN 00352 IPIV( K ) = KP 00353 ELSE 00354 IPIV( K ) = -KP 00355 IPIV( K-1 ) = -KP 00356 END IF 00357 * 00358 * Decrease K and return to the start of the main loop 00359 * 00360 K = K - KSTEP 00361 GO TO 10 00362 * 00363 ELSE 00364 * 00365 * Factorize A as L*D*L**H using the lower triangle of A 00366 * 00367 * K is the main loop index, increasing from 1 to N in steps of 00368 * 1 or 2 00369 * 00370 K = 1 00371 50 CONTINUE 00372 * 00373 * If K > N, exit from loop 00374 * 00375 IF( K.GT.N ) 00376 $ GO TO 90 00377 KSTEP = 1 00378 * 00379 * Determine rows and columns to be interchanged and whether 00380 * a 1-by-1 or 2-by-2 pivot block will be used 00381 * 00382 ABSAKK = ABS( DBLE( A( K, K ) ) ) 00383 * 00384 * IMAX is the row-index of the largest off-diagonal element in 00385 * column K, and COLMAX is its absolute value 00386 * 00387 IF( K.LT.N ) THEN 00388 IMAX = K + IZAMAX( N-K, A( K+1, K ), 1 ) 00389 COLMAX = CABS1( A( IMAX, K ) ) 00390 ELSE 00391 COLMAX = ZERO 00392 END IF 00393 * 00394 IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN 00395 * 00396 * Column K is zero or contains a NaN: set INFO and continue 00397 * 00398 IF( INFO.EQ.0 ) 00399 $ INFO = K 00400 KP = K 00401 A( K, K ) = DBLE( A( K, K ) ) 00402 ELSE 00403 IF( ABSAKK.GE.ALPHA*COLMAX ) THEN 00404 * 00405 * no interchange, use 1-by-1 pivot block 00406 * 00407 KP = K 00408 ELSE 00409 * 00410 * JMAX is the column-index of the largest off-diagonal 00411 * element in row IMAX, and ROWMAX is its absolute value 00412 * 00413 JMAX = K - 1 + IZAMAX( IMAX-K, A( IMAX, K ), LDA ) 00414 ROWMAX = CABS1( A( IMAX, JMAX ) ) 00415 IF( IMAX.LT.N ) THEN 00416 JMAX = IMAX + IZAMAX( N-IMAX, A( IMAX+1, IMAX ), 1 ) 00417 ROWMAX = MAX( ROWMAX, CABS1( A( JMAX, IMAX ) ) ) 00418 END IF 00419 * 00420 IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN 00421 * 00422 * no interchange, use 1-by-1 pivot block 00423 * 00424 KP = K 00425 ELSE IF( ABS( DBLE( A( IMAX, IMAX ) ) ).GE.ALPHA*ROWMAX ) 00426 $ THEN 00427 * 00428 * interchange rows and columns K and IMAX, use 1-by-1 00429 * pivot block 00430 * 00431 KP = IMAX 00432 ELSE 00433 * 00434 * interchange rows and columns K+1 and IMAX, use 2-by-2 00435 * pivot block 00436 * 00437 KP = IMAX 00438 KSTEP = 2 00439 END IF 00440 END IF 00441 * 00442 KK = K + KSTEP - 1 00443 IF( KP.NE.KK ) THEN 00444 * 00445 * Interchange rows and columns KK and KP in the trailing 00446 * submatrix A(k:n,k:n) 00447 * 00448 IF( KP.LT.N ) 00449 $ CALL ZSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 ) 00450 DO 60 J = KK + 1, KP - 1 00451 T = DCONJG( A( J, KK ) ) 00452 A( J, KK ) = DCONJG( A( KP, J ) ) 00453 A( KP, J ) = T 00454 60 CONTINUE 00455 A( KP, KK ) = DCONJG( A( KP, KK ) ) 00456 R1 = DBLE( A( KK, KK ) ) 00457 A( KK, KK ) = DBLE( A( KP, KP ) ) 00458 A( KP, KP ) = R1 00459 IF( KSTEP.EQ.2 ) THEN 00460 A( K, K ) = DBLE( A( K, K ) ) 00461 T = A( K+1, K ) 00462 A( K+1, K ) = A( KP, K ) 00463 A( KP, K ) = T 00464 END IF 00465 ELSE 00466 A( K, K ) = DBLE( A( K, K ) ) 00467 IF( KSTEP.EQ.2 ) 00468 $ A( K+1, K+1 ) = DBLE( A( K+1, K+1 ) ) 00469 END IF 00470 * 00471 * Update the trailing submatrix 00472 * 00473 IF( KSTEP.EQ.1 ) THEN 00474 * 00475 * 1-by-1 pivot block D(k): column k now holds 00476 * 00477 * W(k) = L(k)*D(k) 00478 * 00479 * where L(k) is the k-th column of L 00480 * 00481 IF( K.LT.N ) THEN 00482 * 00483 * Perform a rank-1 update of A(k+1:n,k+1:n) as 00484 * 00485 * A := A - L(k)*D(k)*L(k)**H = A - W(k)*(1/D(k))*W(k)**H 00486 * 00487 R1 = ONE / DBLE( A( K, K ) ) 00488 CALL ZHER( UPLO, N-K, -R1, A( K+1, K ), 1, 00489 $ A( K+1, K+1 ), LDA ) 00490 * 00491 * Store L(k) in column K 00492 * 00493 CALL ZDSCAL( N-K, R1, A( K+1, K ), 1 ) 00494 END IF 00495 ELSE 00496 * 00497 * 2-by-2 pivot block D(k) 00498 * 00499 IF( K.LT.N-1 ) THEN 00500 * 00501 * Perform a rank-2 update of A(k+2:n,k+2:n) as 00502 * 00503 * A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )**H 00504 * = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )**H 00505 * 00506 * where L(k) and L(k+1) are the k-th and (k+1)-th 00507 * columns of L 00508 * 00509 D = DLAPY2( DBLE( A( K+1, K ) ), 00510 $ DIMAG( A( K+1, K ) ) ) 00511 D11 = DBLE( A( K+1, K+1 ) ) / D 00512 D22 = DBLE( A( K, K ) ) / D 00513 TT = ONE / ( D11*D22-ONE ) 00514 D21 = A( K+1, K ) / D 00515 D = TT / D 00516 * 00517 DO 80 J = K + 2, N 00518 WK = D*( D11*A( J, K )-D21*A( J, K+1 ) ) 00519 WKP1 = D*( D22*A( J, K+1 )-DCONJG( D21 )* 00520 $ A( J, K ) ) 00521 DO 70 I = J, N 00522 A( I, J ) = A( I, J ) - A( I, K )*DCONJG( WK ) - 00523 $ A( I, K+1 )*DCONJG( WKP1 ) 00524 70 CONTINUE 00525 A( J, K ) = WK 00526 A( J, K+1 ) = WKP1 00527 A( J, J ) = DCMPLX( DBLE( A( J, J ) ), 0.0D+0 ) 00528 80 CONTINUE 00529 END IF 00530 END IF 00531 END IF 00532 * 00533 * Store details of the interchanges in IPIV 00534 * 00535 IF( KSTEP.EQ.1 ) THEN 00536 IPIV( K ) = KP 00537 ELSE 00538 IPIV( K ) = -KP 00539 IPIV( K+1 ) = -KP 00540 END IF 00541 * 00542 * Increase K and return to the start of the main loop 00543 * 00544 K = K + KSTEP 00545 GO TO 50 00546 * 00547 END IF 00548 * 00549 90 CONTINUE 00550 RETURN 00551 * 00552 * End of ZHETF2 00553 * 00554 END
1.7.3 
