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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 CHETF2( 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 A( LDA, * ) 00015 * .. 00016 * 00017 * Purpose 00018 * ======= 00019 * 00020 * CHETF2 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 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.392 00083 * IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN 00084 * by 00085 * IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. SISNAN(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 REAL ZERO, ONE 00129 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 00130 REAL EIGHT, SEVTEN 00131 PARAMETER ( EIGHT = 8.0E+0, SEVTEN = 17.0E+0 ) 00132 * .. 00133 * .. Local Scalars .. 00134 LOGICAL UPPER 00135 INTEGER I, IMAX, J, JMAX, K, KK, KP, KSTEP 00136 REAL ABSAKK, ALPHA, COLMAX, D, D11, D22, R1, ROWMAX, 00137 $ TT 00138 COMPLEX D12, D21, T, WK, WKM1, WKP1, ZDUM 00139 * .. 00140 * .. External Functions .. 00141 LOGICAL LSAME, SISNAN 00142 INTEGER ICAMAX 00143 REAL SLAPY2 00144 EXTERNAL LSAME, ICAMAX, SLAPY2, SISNAN 00145 * .. 00146 * .. External Subroutines .. 00147 EXTERNAL CHER, CSSCAL, CSWAP, XERBLA 00148 * .. 00149 * .. Intrinsic Functions .. 00150 INTRINSIC ABS, AIMAG, CMPLX, CONJG, MAX, REAL, SQRT 00151 * .. 00152 * .. Statement Functions .. 00153 REAL CABS1 00154 * .. 00155 * .. Statement Function definitions .. 00156 CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( 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( 'CHETF2', -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( REAL( 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 = ICAMAX( 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. SISNAN(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 ) = REAL( 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 + ICAMAX( K-IMAX, A( IMAX, IMAX+1 ), LDA ) 00231 ROWMAX = CABS1( A( IMAX, JMAX ) ) 00232 IF( IMAX.GT.1 ) THEN 00233 JMAX = ICAMAX( 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( REAL( 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 CSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 ) 00266 DO 20 J = KP + 1, KK - 1 00267 T = CONJG( A( J, KK ) ) 00268 A( J, KK ) = CONJG( A( KP, J ) ) 00269 A( KP, J ) = T 00270 20 CONTINUE 00271 A( KP, KK ) = CONJG( A( KP, KK ) ) 00272 R1 = REAL( A( KK, KK ) ) 00273 A( KK, KK ) = REAL( A( KP, KP ) ) 00274 A( KP, KP ) = R1 00275 IF( KSTEP.EQ.2 ) THEN 00276 A( K, K ) = REAL( 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 ) = REAL( A( K, K ) ) 00283 IF( KSTEP.EQ.2 ) 00284 $ A( K-1, K-1 ) = REAL( 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 / REAL( A( K, K ) ) 00302 CALL CHER( UPLO, K-1, -R1, A( 1, K ), 1, A, LDA ) 00303 * 00304 * Store U(k) in column k 00305 * 00306 CALL CSSCAL( 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 = SLAPY2( REAL( A( K-1, K ) ), 00324 $ AIMAG( A( K-1, K ) ) ) 00325 D22 = REAL( A( K-1, K-1 ) ) / D 00326 D11 = REAL( 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 )-CONJG( D12 )*A( J, K ) ) 00333 WK = D*( D22*A( J, K )-D12*A( J, K-1 ) ) 00334 DO 30 I = J, 1, -1 00335 A( I, J ) = A( I, J ) - A( I, K )*CONJG( WK ) - 00336 $ A( I, K-1 )*CONJG( WKM1 ) 00337 30 CONTINUE 00338 A( J, K ) = WK 00339 A( J, K-1 ) = WKM1 00340 A( J, J ) = CMPLX( REAL( A( J, J ) ), 0.0E+0 ) 00341 40 CONTINUE 00342 * 00343 END IF 00344 * 00345 END IF 00346 END IF 00347 * 00348 * Store details of the interchanges in IPIV 00349 * 00350 IF( KSTEP.EQ.1 ) THEN 00351 IPIV( K ) = KP 00352 ELSE 00353 IPIV( K ) = -KP 00354 IPIV( K-1 ) = -KP 00355 END IF 00356 * 00357 * Decrease K and return to the start of the main loop 00358 * 00359 K = K - KSTEP 00360 GO TO 10 00361 * 00362 ELSE 00363 * 00364 * Factorize A as L*D*L**H using the lower triangle of A 00365 * 00366 * K is the main loop index, increasing from 1 to N in steps of 00367 * 1 or 2 00368 * 00369 K = 1 00370 50 CONTINUE 00371 * 00372 * If K > N, exit from loop 00373 * 00374 IF( K.GT.N ) 00375 $ GO TO 90 00376 KSTEP = 1 00377 * 00378 * Determine rows and columns to be interchanged and whether 00379 * a 1-by-1 or 2-by-2 pivot block will be used 00380 * 00381 ABSAKK = ABS( REAL( A( K, K ) ) ) 00382 * 00383 * IMAX is the row-index of the largest off-diagonal element in 00384 * column K, and COLMAX is its absolute value 00385 * 00386 IF( K.LT.N ) THEN 00387 IMAX = K + ICAMAX( N-K, A( K+1, K ), 1 ) 00388 COLMAX = CABS1( A( IMAX, K ) ) 00389 ELSE 00390 COLMAX = ZERO 00391 END IF 00392 * 00393 IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. SISNAN(ABSAKK) ) THEN 00394 * 00395 * Column K is zero or contains a NaN: set INFO and continue 00396 * 00397 IF( INFO.EQ.0 ) 00398 $ INFO = K 00399 KP = K 00400 A( K, K ) = REAL( A( K, K ) ) 00401 ELSE 00402 IF( ABSAKK.GE.ALPHA*COLMAX ) THEN 00403 * 00404 * no interchange, use 1-by-1 pivot block 00405 * 00406 KP = K 00407 ELSE 00408 * 00409 * JMAX is the column-index of the largest off-diagonal 00410 * element in row IMAX, and ROWMAX is its absolute value 00411 * 00412 JMAX = K - 1 + ICAMAX( IMAX-K, A( IMAX, K ), LDA ) 00413 ROWMAX = CABS1( A( IMAX, JMAX ) ) 00414 IF( IMAX.LT.N ) THEN 00415 JMAX = IMAX + ICAMAX( N-IMAX, A( IMAX+1, IMAX ), 1 ) 00416 ROWMAX = MAX( ROWMAX, CABS1( A( JMAX, IMAX ) ) ) 00417 END IF 00418 * 00419 IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN 00420 * 00421 * no interchange, use 1-by-1 pivot block 00422 * 00423 KP = K 00424 ELSE IF( ABS( REAL( A( IMAX, IMAX ) ) ).GE.ALPHA*ROWMAX ) 00425 $ THEN 00426 * 00427 * interchange rows and columns K and IMAX, use 1-by-1 00428 * pivot block 00429 * 00430 KP = IMAX 00431 ELSE 00432 * 00433 * interchange rows and columns K+1 and IMAX, use 2-by-2 00434 * pivot block 00435 * 00436 KP = IMAX 00437 KSTEP = 2 00438 END IF 00439 END IF 00440 * 00441 KK = K + KSTEP - 1 00442 IF( KP.NE.KK ) THEN 00443 * 00444 * Interchange rows and columns KK and KP in the trailing 00445 * submatrix A(k:n,k:n) 00446 * 00447 IF( KP.LT.N ) 00448 $ CALL CSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 ) 00449 DO 60 J = KK + 1, KP - 1 00450 T = CONJG( A( J, KK ) ) 00451 A( J, KK ) = CONJG( A( KP, J ) ) 00452 A( KP, J ) = T 00453 60 CONTINUE 00454 A( KP, KK ) = CONJG( A( KP, KK ) ) 00455 R1 = REAL( A( KK, KK ) ) 00456 A( KK, KK ) = REAL( A( KP, KP ) ) 00457 A( KP, KP ) = R1 00458 IF( KSTEP.EQ.2 ) THEN 00459 A( K, K ) = REAL( A( K, K ) ) 00460 T = A( K+1, K ) 00461 A( K+1, K ) = A( KP, K ) 00462 A( KP, K ) = T 00463 END IF 00464 ELSE 00465 A( K, K ) = REAL( A( K, K ) ) 00466 IF( KSTEP.EQ.2 ) 00467 $ A( K+1, K+1 ) = REAL( A( K+1, K+1 ) ) 00468 END IF 00469 * 00470 * Update the trailing submatrix 00471 * 00472 IF( KSTEP.EQ.1 ) THEN 00473 * 00474 * 1-by-1 pivot block D(k): column k now holds 00475 * 00476 * W(k) = L(k)*D(k) 00477 * 00478 * where L(k) is the k-th column of L 00479 * 00480 IF( K.LT.N ) THEN 00481 * 00482 * Perform a rank-1 update of A(k+1:n,k+1:n) as 00483 * 00484 * A := A - L(k)*D(k)*L(k)**H = A - W(k)*(1/D(k))*W(k)**H 00485 * 00486 R1 = ONE / REAL( A( K, K ) ) 00487 CALL CHER( UPLO, N-K, -R1, A( K+1, K ), 1, 00488 $ A( K+1, K+1 ), LDA ) 00489 * 00490 * Store L(k) in column K 00491 * 00492 CALL CSSCAL( N-K, R1, A( K+1, K ), 1 ) 00493 END IF 00494 ELSE 00495 * 00496 * 2-by-2 pivot block D(k) 00497 * 00498 IF( K.LT.N-1 ) THEN 00499 * 00500 * Perform a rank-2 update of A(k+2:n,k+2:n) as 00501 * 00502 * A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )**H 00503 * = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )**H 00504 * 00505 * where L(k) and L(k+1) are the k-th and (k+1)-th 00506 * columns of L 00507 * 00508 D = SLAPY2( REAL( A( K+1, K ) ), 00509 $ AIMAG( A( K+1, K ) ) ) 00510 D11 = REAL( A( K+1, K+1 ) ) / D 00511 D22 = REAL( A( K, K ) ) / D 00512 TT = ONE / ( D11*D22-ONE ) 00513 D21 = A( K+1, K ) / D 00514 D = TT / D 00515 * 00516 DO 80 J = K + 2, N 00517 WK = D*( D11*A( J, K )-D21*A( J, K+1 ) ) 00518 WKP1 = D*( D22*A( J, K+1 )-CONJG( D21 )*A( J, K ) ) 00519 DO 70 I = J, N 00520 A( I, J ) = A( I, J ) - A( I, K )*CONJG( WK ) - 00521 $ A( I, K+1 )*CONJG( WKP1 ) 00522 70 CONTINUE 00523 A( J, K ) = WK 00524 A( J, K+1 ) = WKP1 00525 A( J, J ) = CMPLX( REAL( A( J, J ) ), 0.0E+0 ) 00526 80 CONTINUE 00527 END IF 00528 END IF 00529 END IF 00530 * 00531 * Store details of the interchanges in IPIV 00532 * 00533 IF( KSTEP.EQ.1 ) THEN 00534 IPIV( K ) = KP 00535 ELSE 00536 IPIV( K ) = -KP 00537 IPIV( K+1 ) = -KP 00538 END IF 00539 * 00540 * Increase K and return to the start of the main loop 00541 * 00542 K = K + KSTEP 00543 GO TO 50 00544 * 00545 END IF 00546 * 00547 90 CONTINUE 00548 RETURN 00549 * 00550 * End of CHETF2 00551 * 00552 END
1.7.3 
