LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ zlatm5()

 subroutine zlatm5 ( integer PRTYPE, integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( ldd, * ) D, integer LDD, complex*16, dimension( lde, * ) E, integer LDE, complex*16, dimension( ldf, * ) F, integer LDF, complex*16, dimension( ldr, * ) R, integer LDR, complex*16, dimension( ldl, * ) L, integer LDL, double precision ALPHA, integer QBLCKA, integer QBLCKB )

ZLATM5

Purpose:
``` ZLATM5 generates matrices involved in the Generalized Sylvester
equation:

A * R - L * B = C
D * R - L * E = F

They also satisfy (the diagonalization condition)

[ I -L ] ( [ A  -C ], [ D -F ] ) [ I  R ] = ( [ A    ], [ D    ] )
[    I ] ( [     B ]  [    E ] ) [    I ]   ( [    B ]  [    E ] )```
Parameters
 [in] PRTYPE ``` PRTYPE is INTEGER "Points" to a certain type of the matrices to generate (see further details).``` [in] M ``` M is INTEGER Specifies the order of A and D and the number of rows in C, F, R and L.``` [in] N ``` N is INTEGER Specifies the order of B and E and the number of columns in C, F, R and L.``` [out] A ``` A is COMPLEX*16 array, dimension (LDA, M). On exit A M-by-M is initialized according to PRTYPE.``` [in] LDA ``` LDA is INTEGER The leading dimension of A.``` [out] B ``` B is COMPLEX*16 array, dimension (LDB, N). On exit B N-by-N is initialized according to PRTYPE.``` [in] LDB ``` LDB is INTEGER The leading dimension of B.``` [out] C ``` C is COMPLEX*16 array, dimension (LDC, N). On exit C M-by-N is initialized according to PRTYPE.``` [in] LDC ``` LDC is INTEGER The leading dimension of C.``` [out] D ``` D is COMPLEX*16 array, dimension (LDD, M). On exit D M-by-M is initialized according to PRTYPE.``` [in] LDD ``` LDD is INTEGER The leading dimension of D.``` [out] E ``` E is COMPLEX*16 array, dimension (LDE, N). On exit E N-by-N is initialized according to PRTYPE.``` [in] LDE ``` LDE is INTEGER The leading dimension of E.``` [out] F ``` F is COMPLEX*16 array, dimension (LDF, N). On exit F M-by-N is initialized according to PRTYPE.``` [in] LDF ``` LDF is INTEGER The leading dimension of F.``` [out] R ``` R is COMPLEX*16 array, dimension (LDR, N). On exit R M-by-N is initialized according to PRTYPE.``` [in] LDR ``` LDR is INTEGER The leading dimension of R.``` [out] L ``` L is COMPLEX*16 array, dimension (LDL, N). On exit L M-by-N is initialized according to PRTYPE.``` [in] LDL ``` LDL is INTEGER The leading dimension of L.``` [in] ALPHA ``` ALPHA is DOUBLE PRECISION Parameter used in generating PRTYPE = 1 and 5 matrices.``` [in] QBLCKA ``` QBLCKA is INTEGER When PRTYPE = 3, specifies the distance between 2-by-2 blocks on the diagonal in A. Otherwise, QBLCKA is not referenced. QBLCKA > 1.``` [in] QBLCKB ``` QBLCKB is INTEGER When PRTYPE = 3, specifies the distance between 2-by-2 blocks on the diagonal in B. Otherwise, QBLCKB is not referenced. QBLCKB > 1.```
Further Details:
```  PRTYPE = 1: A and B are Jordan blocks, D and E are identity matrices

A : if (i == j) then A(i, j) = 1.0
if (j == i + 1) then A(i, j) = -1.0
else A(i, j) = 0.0,            i, j = 1...M

B : if (i == j) then B(i, j) = 1.0 - ALPHA
if (j == i + 1) then B(i, j) = 1.0
else B(i, j) = 0.0,            i, j = 1...N

D : if (i == j) then D(i, j) = 1.0
else D(i, j) = 0.0,            i, j = 1...M

E : if (i == j) then E(i, j) = 1.0
else E(i, j) = 0.0,            i, j = 1...N

L =  R are chosen from [-10...10],
which specifies the right hand sides (C, F).

PRTYPE = 2 or 3: Triangular and/or quasi- triangular.

A : if (i <= j) then A(i, j) = [-1...1]
else A(i, j) = 0.0,             i, j = 1...M

if (PRTYPE = 3) then
A(k + 1, k + 1) = A(k, k)
A(k + 1, k) = [-1...1]
sign(A(k, k + 1) = -(sin(A(k + 1, k))
k = 1, M - 1, QBLCKA

B : if (i <= j) then B(i, j) = [-1...1]
else B(i, j) = 0.0,            i, j = 1...N

if (PRTYPE = 3) then
B(k + 1, k + 1) = B(k, k)
B(k + 1, k) = [-1...1]
sign(B(k, k + 1) = -(sign(B(k + 1, k))
k = 1, N - 1, QBLCKB

D : if (i <= j) then D(i, j) = [-1...1].
else D(i, j) = 0.0,            i, j = 1...M

E : if (i <= j) then D(i, j) = [-1...1]
else E(i, j) = 0.0,            i, j = 1...N

L, R are chosen from [-10...10],
which specifies the right hand sides (C, F).

PRTYPE = 4 Full
A(i, j) = [-10...10]
D(i, j) = [-1...1]    i,j = 1...M
B(i, j) = [-10...10]
E(i, j) = [-1...1]    i,j = 1...N
R(i, j) = [-10...10]
L(i, j) = [-1...1]    i = 1..M ,j = 1...N

L, R specifies the right hand sides (C, F).

PRTYPE = 5 special case common and/or close eigs.```

Definition at line 265 of file zlatm5.f.

268 *
269 * -- LAPACK computational routine --
270 * -- LAPACK is a software package provided by Univ. of Tennessee, --
271 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
272 *
273 * .. Scalar Arguments ..
274  INTEGER LDA, LDB, LDC, LDD, LDE, LDF, LDL, LDR, M, N,
275  \$ PRTYPE, QBLCKA, QBLCKB
276  DOUBLE PRECISION ALPHA
277 * ..
278 * .. Array Arguments ..
279  COMPLEX*16 A( LDA, * ), B( LDB, * ), C( LDC, * ),
280  \$ D( LDD, * ), E( LDE, * ), F( LDF, * ),
281  \$ L( LDL, * ), R( LDR, * )
282 * ..
283 *
284 * =====================================================================
285 *
286 * .. Parameters ..
287  COMPLEX*16 ONE, TWO, ZERO, HALF, TWENTY
288  parameter( one = ( 1.0d+0, 0.0d+0 ),
289  \$ two = ( 2.0d+0, 0.0d+0 ),
290  \$ zero = ( 0.0d+0, 0.0d+0 ),
291  \$ half = ( 0.5d+0, 0.0d+0 ),
292  \$ twenty = ( 2.0d+1, 0.0d+0 ) )
293 * ..
294 * .. Local Scalars ..
295  INTEGER I, J, K
296  COMPLEX*16 IMEPS, REEPS
297 * ..
298 * .. Intrinsic Functions ..
299  INTRINSIC dcmplx, mod, sin
300 * ..
301 * .. External Subroutines ..
302  EXTERNAL zgemm
303 * ..
304 * .. Executable Statements ..
305 *
306  IF( prtype.EQ.1 ) THEN
307  DO 20 i = 1, m
308  DO 10 j = 1, m
309  IF( i.EQ.j ) THEN
310  a( i, j ) = one
311  d( i, j ) = one
312  ELSE IF( i.EQ.j-1 ) THEN
313  a( i, j ) = -one
314  d( i, j ) = zero
315  ELSE
316  a( i, j ) = zero
317  d( i, j ) = zero
318  END IF
319  10 CONTINUE
320  20 CONTINUE
321 *
322  DO 40 i = 1, n
323  DO 30 j = 1, n
324  IF( i.EQ.j ) THEN
325  b( i, j ) = one - alpha
326  e( i, j ) = one
327  ELSE IF( i.EQ.j-1 ) THEN
328  b( i, j ) = one
329  e( i, j ) = zero
330  ELSE
331  b( i, j ) = zero
332  e( i, j ) = zero
333  END IF
334  30 CONTINUE
335  40 CONTINUE
336 *
337  DO 60 i = 1, m
338  DO 50 j = 1, n
339  r( i, j ) = ( half-sin( dcmplx( i / j ) ) )*twenty
340  l( i, j ) = r( i, j )
341  50 CONTINUE
342  60 CONTINUE
343 *
344  ELSE IF( prtype.EQ.2 .OR. prtype.EQ.3 ) THEN
345  DO 80 i = 1, m
346  DO 70 j = 1, m
347  IF( i.LE.j ) THEN
348  a( i, j ) = ( half-sin( dcmplx( i ) ) )*two
349  d( i, j ) = ( half-sin( dcmplx( i*j ) ) )*two
350  ELSE
351  a( i, j ) = zero
352  d( i, j ) = zero
353  END IF
354  70 CONTINUE
355  80 CONTINUE
356 *
357  DO 100 i = 1, n
358  DO 90 j = 1, n
359  IF( i.LE.j ) THEN
360  b( i, j ) = ( half-sin( dcmplx( i+j ) ) )*two
361  e( i, j ) = ( half-sin( dcmplx( j ) ) )*two
362  ELSE
363  b( i, j ) = zero
364  e( i, j ) = zero
365  END IF
366  90 CONTINUE
367  100 CONTINUE
368 *
369  DO 120 i = 1, m
370  DO 110 j = 1, n
371  r( i, j ) = ( half-sin( dcmplx( i*j ) ) )*twenty
372  l( i, j ) = ( half-sin( dcmplx( i+j ) ) )*twenty
373  110 CONTINUE
374  120 CONTINUE
375 *
376  IF( prtype.EQ.3 ) THEN
377  IF( qblcka.LE.1 )
378  \$ qblcka = 2
379  DO 130 k = 1, m - 1, qblcka
380  a( k+1, k+1 ) = a( k, k )
381  a( k+1, k ) = -sin( a( k, k+1 ) )
382  130 CONTINUE
383 *
384  IF( qblckb.LE.1 )
385  \$ qblckb = 2
386  DO 140 k = 1, n - 1, qblckb
387  b( k+1, k+1 ) = b( k, k )
388  b( k+1, k ) = -sin( b( k, k+1 ) )
389  140 CONTINUE
390  END IF
391 *
392  ELSE IF( prtype.EQ.4 ) THEN
393  DO 160 i = 1, m
394  DO 150 j = 1, m
395  a( i, j ) = ( half-sin( dcmplx( i*j ) ) )*twenty
396  d( i, j ) = ( half-sin( dcmplx( i+j ) ) )*two
397  150 CONTINUE
398  160 CONTINUE
399 *
400  DO 180 i = 1, n
401  DO 170 j = 1, n
402  b( i, j ) = ( half-sin( dcmplx( i+j ) ) )*twenty
403  e( i, j ) = ( half-sin( dcmplx( i*j ) ) )*two
404  170 CONTINUE
405  180 CONTINUE
406 *
407  DO 200 i = 1, m
408  DO 190 j = 1, n
409  r( i, j ) = ( half-sin( dcmplx( j / i ) ) )*twenty
410  l( i, j ) = ( half-sin( dcmplx( i*j ) ) )*two
411  190 CONTINUE
412  200 CONTINUE
413 *
414  ELSE IF( prtype.GE.5 ) THEN
415  reeps = half*two*twenty / alpha
416  imeps = ( half-two ) / alpha
417  DO 220 i = 1, m
418  DO 210 j = 1, n
419  r( i, j ) = ( half-sin( dcmplx( i*j ) ) )*alpha / twenty
420  l( i, j ) = ( half-sin( dcmplx( i+j ) ) )*alpha / twenty
421  210 CONTINUE
422  220 CONTINUE
423 *
424  DO 230 i = 1, m
425  d( i, i ) = one
426  230 CONTINUE
427 *
428  DO 240 i = 1, m
429  IF( i.LE.4 ) THEN
430  a( i, i ) = one
431  IF( i.GT.2 )
432  \$ a( i, i ) = one + reeps
433  IF( mod( i, 2 ).NE.0 .AND. i.LT.m ) THEN
434  a( i, i+1 ) = imeps
435  ELSE IF( i.GT.1 ) THEN
436  a( i, i-1 ) = -imeps
437  END IF
438  ELSE IF( i.LE.8 ) THEN
439  IF( i.LE.6 ) THEN
440  a( i, i ) = reeps
441  ELSE
442  a( i, i ) = -reeps
443  END IF
444  IF( mod( i, 2 ).NE.0 .AND. i.LT.m ) THEN
445  a( i, i+1 ) = one
446  ELSE IF( i.GT.1 ) THEN
447  a( i, i-1 ) = -one
448  END IF
449  ELSE
450  a( i, i ) = one
451  IF( mod( i, 2 ).NE.0 .AND. i.LT.m ) THEN
452  a( i, i+1 ) = imeps*2
453  ELSE IF( i.GT.1 ) THEN
454  a( i, i-1 ) = -imeps*2
455  END IF
456  END IF
457  240 CONTINUE
458 *
459  DO 250 i = 1, n
460  e( i, i ) = one
461  IF( i.LE.4 ) THEN
462  b( i, i ) = -one
463  IF( i.GT.2 )
464  \$ b( i, i ) = one - reeps
465  IF( mod( i, 2 ).NE.0 .AND. i.LT.n ) THEN
466  b( i, i+1 ) = imeps
467  ELSE IF( i.GT.1 ) THEN
468  b( i, i-1 ) = -imeps
469  END IF
470  ELSE IF( i.LE.8 ) THEN
471  IF( i.LE.6 ) THEN
472  b( i, i ) = reeps
473  ELSE
474  b( i, i ) = -reeps
475  END IF
476  IF( mod( i, 2 ).NE.0 .AND. i.LT.n ) THEN
477  b( i, i+1 ) = one + imeps
478  ELSE IF( i.GT.1 ) THEN
479  b( i, i-1 ) = -one - imeps
480  END IF
481  ELSE
482  b( i, i ) = one - reeps
483  IF( mod( i, 2 ).NE.0 .AND. i.LT.n ) THEN
484  b( i, i+1 ) = imeps*2
485  ELSE IF( i.GT.1 ) THEN
486  b( i, i-1 ) = -imeps*2
487  END IF
488  END IF
489  250 CONTINUE
490  END IF
491 *
492 * Compute rhs (C, F)
493 *
494  CALL zgemm( 'N', 'N', m, n, m, one, a, lda, r, ldr, zero, c, ldc )
495  CALL zgemm( 'N', 'N', m, n, n, -one, l, ldl, b, ldb, one, c, ldc )
496  CALL zgemm( 'N', 'N', m, n, m, one, d, ldd, r, ldr, zero, f, ldf )
497  CALL zgemm( 'N', 'N', m, n, n, -one, l, ldl, e, lde, one, f, ldf )
498 *
499 * End of ZLATM5
500 *
logical function lde(RI, RJ, LR)
Definition: dblat2.f:2942
subroutine zgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
ZGEMM
Definition: zgemm.f:187
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