xref: /petsc/include/petscmath.h (revision 6497c311e7b976d467be1503c1effce92a60525c)

/*
    PETSc mathematics include file. Defines certain basic mathematical
    constants and functions for working with single, double, and quad precision
    floating point numbers as well as complex single and double.

    This file is included by petscsys.h and should not be used directly.
*/
#pragma once

#include <math.h>
#include <petscmacros.h>
#include <petscsystypes.h>

/* SUBMANSEC = Sys */

/*
   Defines operations that are different for complex and real numbers.
   All PETSc objects in one program are built around the object
   PetscScalar which is either always a real or a complex.
*/

/*
    Real number definitions
 */
#if defined(PETSC_USE_REAL_SINGLE)
  #define PetscSqrtReal(a)        sqrtf(a)
  #define PetscCbrtReal(a)        cbrtf(a)
  #define PetscHypotReal(a, b)    hypotf(a, b)
  #define PetscAtan2Real(a, b)    atan2f(a, b)
  #define PetscPowReal(a, b)      powf(a, b)
  #define PetscExpReal(a)         expf(a)
  #define PetscLogReal(a)         logf(a)
  #define PetscLog10Real(a)       log10f(a)
  #define PetscLog2Real(a)        log2f(a)
  #define PetscSinReal(a)         sinf(a)
  #define PetscCosReal(a)         cosf(a)
  #define PetscTanReal(a)         tanf(a)
  #define PetscAsinReal(a)        asinf(a)
  #define PetscAcosReal(a)        acosf(a)
  #define PetscAtanReal(a)        atanf(a)
  #define PetscSinhReal(a)        sinhf(a)
  #define PetscCoshReal(a)        coshf(a)
  #define PetscTanhReal(a)        tanhf(a)
  #define PetscAsinhReal(a)       asinhf(a)
  #define PetscAcoshReal(a)       acoshf(a)
  #define PetscAtanhReal(a)       atanhf(a)
  #define PetscErfReal(a)         erff(a)
  #define PetscCeilReal(a)        ceilf(a)
  #define PetscFloorReal(a)       floorf(a)
  #define PetscFmodReal(a, b)     fmodf(a, b)
  #define PetscCopysignReal(a, b) copysignf(a, b)
  #define PetscTGamma(a)          tgammaf(a)
  #if defined(PETSC_HAVE_LGAMMA_IS_GAMMA)
    #define PetscLGamma(a) gammaf(a)
  #else
    #define PetscLGamma(a) lgammaf(a)
  #endif

#elif defined(PETSC_USE_REAL_DOUBLE)
  #define PetscSqrtReal(a)        sqrt(a)
  #define PetscCbrtReal(a)        cbrt(a)
  #define PetscHypotReal(a, b)    hypot(a, b)
  #define PetscAtan2Real(a, b)    atan2(a, b)
  #define PetscPowReal(a, b)      pow(a, b)
  #define PetscExpReal(a)         exp(a)
  #define PetscLogReal(a)         log(a)
  #define PetscLog10Real(a)       log10(a)
  #define PetscLog2Real(a)        log2(a)
  #define PetscSinReal(a)         sin(a)
  #define PetscCosReal(a)         cos(a)
  #define PetscTanReal(a)         tan(a)
  #define PetscAsinReal(a)        asin(a)
  #define PetscAcosReal(a)        acos(a)
  #define PetscAtanReal(a)        atan(a)
  #define PetscSinhReal(a)        sinh(a)
  #define PetscCoshReal(a)        cosh(a)
  #define PetscTanhReal(a)        tanh(a)
  #define PetscAsinhReal(a)       asinh(a)
  #define PetscAcoshReal(a)       acosh(a)
  #define PetscAtanhReal(a)       atanh(a)
  #define PetscErfReal(a)         erf(a)
  #define PetscCeilReal(a)        ceil(a)
  #define PetscFloorReal(a)       floor(a)
  #define PetscFmodReal(a, b)     fmod(a, b)
  #define PetscCopysignReal(a, b) copysign(a, b)
  #define PetscTGamma(a)          tgamma(a)
  #if defined(PETSC_HAVE_LGAMMA_IS_GAMMA)
    #define PetscLGamma(a) gamma(a)
  #else
    #define PetscLGamma(a) lgamma(a)
  #endif

#elif defined(PETSC_USE_REAL___FLOAT128)
  #define PetscSqrtReal(a)        sqrtq(a)
  #define PetscCbrtReal(a)        cbrtq(a)
  #define PetscHypotReal(a, b)    hypotq(a, b)
  #define PetscAtan2Real(a, b)    atan2q(a, b)
  #define PetscPowReal(a, b)      powq(a, b)
  #define PetscExpReal(a)         expq(a)
  #define PetscLogReal(a)         logq(a)
  #define PetscLog10Real(a)       log10q(a)
  #define PetscLog2Real(a)        log2q(a)
  #define PetscSinReal(a)         sinq(a)
  #define PetscCosReal(a)         cosq(a)
  #define PetscTanReal(a)         tanq(a)
  #define PetscAsinReal(a)        asinq(a)
  #define PetscAcosReal(a)        acosq(a)
  #define PetscAtanReal(a)        atanq(a)
  #define PetscSinhReal(a)        sinhq(a)
  #define PetscCoshReal(a)        coshq(a)
  #define PetscTanhReal(a)        tanhq(a)
  #define PetscAsinhReal(a)       asinhq(a)
  #define PetscAcoshReal(a)       acoshq(a)
  #define PetscAtanhReal(a)       atanhq(a)
  #define PetscErfReal(a)         erfq(a)
  #define PetscCeilReal(a)        ceilq(a)
  #define PetscFloorReal(a)       floorq(a)
  #define PetscFmodReal(a, b)     fmodq(a, b)
  #define PetscCopysignReal(a, b) copysignq(a, b)
  #define PetscTGamma(a)          tgammaq(a)
  #if defined(PETSC_HAVE_LGAMMA_IS_GAMMA)
    #define PetscLGamma(a) gammaq(a)
  #else
    #define PetscLGamma(a) lgammaq(a)
  #endif

#elif defined(PETSC_USE_REAL___FP16)
  #define PetscSqrtReal(a)        sqrtf(a)
  #define PetscCbrtReal(a)        cbrtf(a)
  #define PetscHypotReal(a, b)    hypotf(a, b)
  #define PetscAtan2Real(a, b)    atan2f(a, b)
  #define PetscPowReal(a, b)      powf(a, b)
  #define PetscExpReal(a)         expf(a)
  #define PetscLogReal(a)         logf(a)
  #define PetscLog10Real(a)       log10f(a)
  #define PetscLog2Real(a)        log2f(a)
  #define PetscSinReal(a)         sinf(a)
  #define PetscCosReal(a)         cosf(a)
  #define PetscTanReal(a)         tanf(a)
  #define PetscAsinReal(a)        asinf(a)
  #define PetscAcosReal(a)        acosf(a)
  #define PetscAtanReal(a)        atanf(a)
  #define PetscSinhReal(a)        sinhf(a)
  #define PetscCoshReal(a)        coshf(a)
  #define PetscTanhReal(a)        tanhf(a)
  #define PetscAsinhReal(a)       asinhf(a)
  #define PetscAcoshReal(a)       acoshf(a)
  #define PetscAtanhReal(a)       atanhf(a)
  #define PetscErfReal(a)         erff(a)
  #define PetscCeilReal(a)        ceilf(a)
  #define PetscFloorReal(a)       floorf(a)
  #define PetscFmodReal(a, b)     fmodf(a, b)
  #define PetscCopysignReal(a, b) copysignf(a, b)
  #define PetscTGamma(a)          tgammaf(a)
  #if defined(PETSC_HAVE_LGAMMA_IS_GAMMA)
    #define PetscLGamma(a) gammaf(a)
  #else
    #define PetscLGamma(a) lgammaf(a)
  #endif

#endif /* PETSC_USE_REAL_* */

static inline PetscReal PetscSignReal(PetscReal a)
{
  return (PetscReal)((a < (PetscReal)0) ? -1 : ((a > (PetscReal)0) ? 1 : 0));
}

#if !defined(PETSC_HAVE_LOG2)
  #undef PetscLog2Real
static inline PetscReal PetscLog2Real(PetscReal a)
{
  return PetscLogReal(a) / PetscLogReal((PetscReal)2);
}
#endif

#if defined(PETSC_HAVE_REAL___FLOAT128) && !defined(PETSC_SKIP_REAL___FLOAT128)
PETSC_EXTERN MPI_Datatype MPIU___FLOAT128 PETSC_ATTRIBUTE_MPI_TYPE_TAG(__float128);
#endif
#if defined(PETSC_HAVE_REAL___FP16) && !defined(PETSC_SKIP_REAL___FP16)
PETSC_EXTERN MPI_Datatype MPIU___FP16 PETSC_ATTRIBUTE_MPI_TYPE_TAG(__fp16);
#endif

/*MC
   MPIU_REAL - Portable MPI datatype corresponding to `PetscReal` independent of what precision `PetscReal` is in

   Level: beginner

   Note:
   In MPI calls that require an MPI datatype that matches a `PetscReal` or array of `PetscReal` values, pass this value.

.seealso: `PetscReal`, `PetscScalar`, `PetscComplex`, `PetscInt`, `MPIU_SCALAR`, `MPIU_COMPLEX`, `MPIU_INT`
M*/
#if defined(PETSC_USE_REAL_SINGLE)
  #define MPIU_REAL MPI_FLOAT
#elif defined(PETSC_USE_REAL_DOUBLE)
  #define MPIU_REAL MPI_DOUBLE
#elif defined(PETSC_USE_REAL___FLOAT128)
  #define MPIU_REAL MPIU___FLOAT128
#elif defined(PETSC_USE_REAL___FP16)
  #define MPIU_REAL MPIU___FP16
#endif /* PETSC_USE_REAL_* */

/*
    Complex number definitions
 */
#if defined(PETSC_HAVE_COMPLEX)
  #if defined(__cplusplus) && !defined(PETSC_USE_REAL___FLOAT128)
  /* C++ support of complex number */

    #define PetscRealPartComplex(a)      (static_cast<PetscComplex>(a)).real()
    #define PetscImaginaryPartComplex(a) (static_cast<PetscComplex>(a)).imag()
    #define PetscAbsComplex(a)           petsccomplexlib::abs(static_cast<PetscComplex>(a))
    #define PetscArgComplex(a)           petsccomplexlib::arg(static_cast<PetscComplex>(a))
    #define PetscConjComplex(a)          petsccomplexlib::conj(static_cast<PetscComplex>(a))
    #define PetscSqrtComplex(a)          petsccomplexlib::sqrt(static_cast<PetscComplex>(a))
    #define PetscPowComplex(a, b)        petsccomplexlib::pow(static_cast<PetscComplex>(a), static_cast<PetscComplex>(b))
    #define PetscExpComplex(a)           petsccomplexlib::exp(static_cast<PetscComplex>(a))
    #define PetscLogComplex(a)           petsccomplexlib::log(static_cast<PetscComplex>(a))
    #define PetscSinComplex(a)           petsccomplexlib::sin(static_cast<PetscComplex>(a))
    #define PetscCosComplex(a)           petsccomplexlib::cos(static_cast<PetscComplex>(a))
    #define PetscTanComplex(a)           petsccomplexlib::tan(static_cast<PetscComplex>(a))
    #define PetscAsinComplex(a)          petsccomplexlib::asin(static_cast<PetscComplex>(a))
    #define PetscAcosComplex(a)          petsccomplexlib::acos(static_cast<PetscComplex>(a))
    #define PetscAtanComplex(a)          petsccomplexlib::atan(static_cast<PetscComplex>(a))
    #define PetscSinhComplex(a)          petsccomplexlib::sinh(static_cast<PetscComplex>(a))
    #define PetscCoshComplex(a)          petsccomplexlib::cosh(static_cast<PetscComplex>(a))
    #define PetscTanhComplex(a)          petsccomplexlib::tanh(static_cast<PetscComplex>(a))
    #define PetscAsinhComplex(a)         petsccomplexlib::asinh(static_cast<PetscComplex>(a))
    #define PetscAcoshComplex(a)         petsccomplexlib::acosh(static_cast<PetscComplex>(a))
    #define PetscAtanhComplex(a)         petsccomplexlib::atanh(static_cast<PetscComplex>(a))

  /* TODO: Add configure tests

#if !defined(PETSC_HAVE_CXX_TAN_COMPLEX)
#undef PetscTanComplex
static inline PetscComplex PetscTanComplex(PetscComplex z)
{
  return PetscSinComplex(z)/PetscCosComplex(z);
}
#endif

#if !defined(PETSC_HAVE_CXX_TANH_COMPLEX)
#undef PetscTanhComplex
static inline PetscComplex PetscTanhComplex(PetscComplex z)
{
  return PetscSinhComplex(z)/PetscCoshComplex(z);
}
#endif

#if !defined(PETSC_HAVE_CXX_ASIN_COMPLEX)
#undef PetscAsinComplex
static inline PetscComplex PetscAsinComplex(PetscComplex z)
{
  const PetscComplex j(0,1);
  return -j*PetscLogComplex(j*z+PetscSqrtComplex(1.0f-z*z));
}
#endif

#if !defined(PETSC_HAVE_CXX_ACOS_COMPLEX)
#undef PetscAcosComplex
static inline PetscComplex PetscAcosComplex(PetscComplex z)
{
  const PetscComplex j(0,1);
  return j*PetscLogComplex(z-j*PetscSqrtComplex(1.0f-z*z));
}
#endif

#if !defined(PETSC_HAVE_CXX_ATAN_COMPLEX)
#undef PetscAtanComplex
static inline PetscComplex PetscAtanComplex(PetscComplex z)
{
  const PetscComplex j(0,1);
  return 0.5f*j*PetscLogComplex((1.0f-j*z)/(1.0f+j*z));
}
#endif

#if !defined(PETSC_HAVE_CXX_ASINH_COMPLEX)
#undef PetscAsinhComplex
static inline PetscComplex PetscAsinhComplex(PetscComplex z)
{
  return PetscLogComplex(z+PetscSqrtComplex(z*z+1.0f));
}
#endif

#if !defined(PETSC_HAVE_CXX_ACOSH_COMPLEX)
#undef PetscAcoshComplex
static inline PetscComplex PetscAcoshComplex(PetscComplex z)
{
  return PetscLogComplex(z+PetscSqrtComplex(z*z-1.0f));
}
#endif

#if !defined(PETSC_HAVE_CXX_ATANH_COMPLEX)
#undef PetscAtanhComplex
static inline PetscComplex PetscAtanhComplex(PetscComplex z)
{
  return 0.5f*PetscLogComplex((1.0f+z)/(1.0f-z));
}
#endif

*/

  #else /* C99 support of complex number */

    #if defined(PETSC_USE_REAL_SINGLE)
      #define PetscRealPartComplex(a)      crealf(a)
      #define PetscImaginaryPartComplex(a) cimagf(a)
      #define PetscAbsComplex(a)           cabsf(a)
      #define PetscArgComplex(a)           cargf(a)
      #define PetscConjComplex(a)          conjf(a)
      #define PetscSqrtComplex(a)          csqrtf(a)
      #define PetscPowComplex(a, b)        cpowf(a, b)
      #define PetscExpComplex(a)           cexpf(a)
      #define PetscLogComplex(a)           clogf(a)
      #define PetscSinComplex(a)           csinf(a)
      #define PetscCosComplex(a)           ccosf(a)
      #define PetscTanComplex(a)           ctanf(a)
      #define PetscAsinComplex(a)          casinf(a)
      #define PetscAcosComplex(a)          cacosf(a)
      #define PetscAtanComplex(a)          catanf(a)
      #define PetscSinhComplex(a)          csinhf(a)
      #define PetscCoshComplex(a)          ccoshf(a)
      #define PetscTanhComplex(a)          ctanhf(a)
      #define PetscAsinhComplex(a)         casinhf(a)
      #define PetscAcoshComplex(a)         cacoshf(a)
      #define PetscAtanhComplex(a)         catanhf(a)

    #elif defined(PETSC_USE_REAL_DOUBLE)
      #define PetscRealPartComplex(a)      creal(a)
      #define PetscImaginaryPartComplex(a) cimag(a)
      #define PetscAbsComplex(a)           cabs(a)
      #define PetscArgComplex(a)           carg(a)
      #define PetscConjComplex(a)          conj(a)
      #define PetscSqrtComplex(a)          csqrt(a)
      #define PetscPowComplex(a, b)        cpow(a, b)
      #define PetscExpComplex(a)           cexp(a)
      #define PetscLogComplex(a)           clog(a)
      #define PetscSinComplex(a)           csin(a)
      #define PetscCosComplex(a)           ccos(a)
      #define PetscTanComplex(a)           ctan(a)
      #define PetscAsinComplex(a)          casin(a)
      #define PetscAcosComplex(a)          cacos(a)
      #define PetscAtanComplex(a)          catan(a)
      #define PetscSinhComplex(a)          csinh(a)
      #define PetscCoshComplex(a)          ccosh(a)
      #define PetscTanhComplex(a)          ctanh(a)
      #define PetscAsinhComplex(a)         casinh(a)
      #define PetscAcoshComplex(a)         cacosh(a)
      #define PetscAtanhComplex(a)         catanh(a)

    #elif defined(PETSC_USE_REAL___FLOAT128)
      #define PetscRealPartComplex(a)      crealq(a)
      #define PetscImaginaryPartComplex(a) cimagq(a)
      #define PetscAbsComplex(a)           cabsq(a)
      #define PetscArgComplex(a)           cargq(a)
      #define PetscConjComplex(a)          conjq(a)
      #define PetscSqrtComplex(a)          csqrtq(a)
      #define PetscPowComplex(a, b)        cpowq(a, b)
      #define PetscExpComplex(a)           cexpq(a)
      #define PetscLogComplex(a)           clogq(a)
      #define PetscSinComplex(a)           csinq(a)
      #define PetscCosComplex(a)           ccosq(a)
      #define PetscTanComplex(a)           ctanq(a)
      #define PetscAsinComplex(a)          casinq(a)
      #define PetscAcosComplex(a)          cacosq(a)
      #define PetscAtanComplex(a)          catanq(a)
      #define PetscSinhComplex(a)          csinhq(a)
      #define PetscCoshComplex(a)          ccoshq(a)
      #define PetscTanhComplex(a)          ctanhq(a)
      #define PetscAsinhComplex(a)         casinhq(a)
      #define PetscAcoshComplex(a)         cacoshq(a)
      #define PetscAtanhComplex(a)         catanhq(a)

    #endif /* PETSC_USE_REAL_* */
  #endif   /* (__cplusplus) */

/*MC
   PETSC_i - the pure imaginary complex number i

   Level: intermediate

.seealso: `PetscComplex`, `PetscScalar`
M*/
PETSC_EXTERN PetscComplex PETSC_i;

/*
   Try to do the right thing for complex number construction: see
   http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1464.htm
   for details
*/
static inline PetscComplex PetscCMPLX(PetscReal x, PetscReal y)
{
  #if defined(__cplusplus) && !defined(PETSC_USE_REAL___FLOAT128)
  return PetscComplex(x, y);
  #elif defined(_Imaginary_I)
  return x + y * _Imaginary_I;
  #else
  { /* In both C99 and C11 (ISO/IEC 9899, Section 6.2.5),

       "For each floating type there is a corresponding real type, which is always a real floating
       type. For real floating types, it is the same type. For complex types, it is the type given
       by deleting the keyword _Complex from the type name."

       So type punning should be portable. */
    union
    {
      PetscComplex z;
      PetscReal    f[2];
    } uz;

    uz.f[0] = x;
    uz.f[1] = y;
    return uz.z;
  }
  #endif
}

  #define MPIU_C_COMPLEX        MPI_C_COMPLEX PETSC_DEPRECATED_MACRO(3, 15, 0, "MPI_C_COMPLEX", )
  #define MPIU_C_DOUBLE_COMPLEX MPI_C_DOUBLE_COMPLEX PETSC_DEPRECATED_MACRO(3, 15, 0, "MPI_C_DOUBLE_COMPLEX", )

  #if defined(PETSC_HAVE_REAL___FLOAT128) && !defined(PETSC_SKIP_REAL___FLOAT128)
    // if complex is not used, then quadmath.h won't be included by petscsystypes.h
    #if defined(PETSC_USE_COMPLEX)
      #define MPIU___COMPLEX128_ATTR_TAG PETSC_ATTRIBUTE_MPI_TYPE_TAG(__complex128)
    #else
      #define MPIU___COMPLEX128_ATTR_TAG
    #endif

PETSC_EXTERN MPI_Datatype MPIU___COMPLEX128 MPIU___COMPLEX128_ATTR_TAG;

    #undef MPIU___COMPLEX128_ATTR_TAG
  #endif /* PETSC_HAVE_REAL___FLOAT128 */

  /*MC
   MPIU_COMPLEX - Portable MPI datatype corresponding to `PetscComplex` independent of the precision of `PetscComplex`

   Level: beginner

   Note:
   In MPI calls that require an MPI datatype that matches a `PetscComplex` or array of `PetscComplex` values, pass this value.

.seealso: `PetscReal`, `PetscScalar`, `PetscComplex`, `PetscInt`, `MPIU_REAL`, `MPIU_SCALAR`, `MPIU_COMPLEX`, `MPIU_INT`, `PETSC_i`
M*/
  #if defined(PETSC_USE_REAL_SINGLE)
    #define MPIU_COMPLEX MPI_C_COMPLEX
  #elif defined(PETSC_USE_REAL_DOUBLE)
    #define MPIU_COMPLEX MPI_C_DOUBLE_COMPLEX
  #elif defined(PETSC_USE_REAL___FLOAT128)
    #define MPIU_COMPLEX MPIU___COMPLEX128
  #elif defined(PETSC_USE_REAL___FP16)
    #define MPIU_COMPLEX MPI_C_COMPLEX
  #endif /* PETSC_USE_REAL_* */

#endif /* PETSC_HAVE_COMPLEX */

/*
    Scalar number definitions
 */
#if defined(PETSC_USE_COMPLEX) && defined(PETSC_HAVE_COMPLEX)
  /*MC
   MPIU_SCALAR - Portable MPI datatype corresponding to `PetscScalar` independent of the precision of `PetscScalar`

   Level: beginner

   Note:
   In MPI calls that require an MPI datatype that matches a `PetscScalar` or array of `PetscScalar` values, pass this value.

.seealso: `PetscReal`, `PetscScalar`, `PetscComplex`, `PetscInt`, `MPIU_REAL`, `MPIU_COMPLEX`, `MPIU_INT`
M*/
  #define MPIU_SCALAR MPIU_COMPLEX

  /*MC
   PetscRealPart - Returns the real part of a `PetscScalar`

   Synopsis:
   #include <petscmath.h>
   PetscReal PetscRealPart(PetscScalar v)

   Not Collective

   Input Parameter:
.  v - value to find the real part of

   Level: beginner

.seealso: `PetscScalar`, `PetscImaginaryPart()`, `PetscMax()`, `PetscClipInterval()`, `PetscAbsInt()`, `PetscAbsReal()`, `PetscSqr()`
M*/
  #define PetscRealPart(a) PetscRealPartComplex(a)

  /*MC
   PetscImaginaryPart - Returns the imaginary part of a `PetscScalar`

   Synopsis:
   #include <petscmath.h>
   PetscReal PetscImaginaryPart(PetscScalar v)

   Not Collective

   Input Parameter:
.  v - value to find the imaginary part of

   Level: beginner

   Note:
   If PETSc was configured for real numbers then this always returns the value 0

.seealso: `PetscScalar`, `PetscRealPart()`, `PetscMax()`, `PetscClipInterval()`, `PetscAbsInt()`, `PetscAbsReal()`, `PetscSqr()`
M*/
  #define PetscImaginaryPart(a) PetscImaginaryPartComplex(a)

  #define PetscAbsScalar(a)    PetscAbsComplex(a)
  #define PetscArgScalar(a)    PetscArgComplex(a)
  #define PetscConj(a)         PetscConjComplex(a)
  #define PetscSqrtScalar(a)   PetscSqrtComplex(a)
  #define PetscPowScalar(a, b) PetscPowComplex(a, b)
  #define PetscExpScalar(a)    PetscExpComplex(a)
  #define PetscLogScalar(a)    PetscLogComplex(a)
  #define PetscSinScalar(a)    PetscSinComplex(a)
  #define PetscCosScalar(a)    PetscCosComplex(a)
  #define PetscTanScalar(a)    PetscTanComplex(a)
  #define PetscAsinScalar(a)   PetscAsinComplex(a)
  #define PetscAcosScalar(a)   PetscAcosComplex(a)
  #define PetscAtanScalar(a)   PetscAtanComplex(a)
  #define PetscSinhScalar(a)   PetscSinhComplex(a)
  #define PetscCoshScalar(a)   PetscCoshComplex(a)
  #define PetscTanhScalar(a)   PetscTanhComplex(a)
  #define PetscAsinhScalar(a)  PetscAsinhComplex(a)
  #define PetscAcoshScalar(a)  PetscAcoshComplex(a)
  #define PetscAtanhScalar(a)  PetscAtanhComplex(a)

#else /* PETSC_USE_COMPLEX */
  #define MPIU_SCALAR           MPIU_REAL
  #define PetscRealPart(a)      (a)
  #define PetscImaginaryPart(a) ((PetscReal)0)
  #define PetscAbsScalar(a)     PetscAbsReal(a)
  #define PetscArgScalar(a)     (((a) < (PetscReal)0) ? PETSC_PI : (PetscReal)0)
  #define PetscConj(a)          (a)
  #define PetscSqrtScalar(a)    PetscSqrtReal(a)
  #define PetscPowScalar(a, b)  PetscPowReal(a, b)
  #define PetscExpScalar(a)     PetscExpReal(a)
  #define PetscLogScalar(a)     PetscLogReal(a)
  #define PetscSinScalar(a)     PetscSinReal(a)
  #define PetscCosScalar(a)     PetscCosReal(a)
  #define PetscTanScalar(a)     PetscTanReal(a)
  #define PetscAsinScalar(a)    PetscAsinReal(a)
  #define PetscAcosScalar(a)    PetscAcosReal(a)
  #define PetscAtanScalar(a)    PetscAtanReal(a)
  #define PetscSinhScalar(a)    PetscSinhReal(a)
  #define PetscCoshScalar(a)    PetscCoshReal(a)
  #define PetscTanhScalar(a)    PetscTanhReal(a)
  #define PetscAsinhScalar(a)   PetscAsinhReal(a)
  #define PetscAcoshScalar(a)   PetscAcoshReal(a)
  #define PetscAtanhScalar(a)   PetscAtanhReal(a)

#endif /* PETSC_USE_COMPLEX */

/*
   Certain objects may be created using either single or double precision.
   This is currently not used.
*/
typedef enum {
  PETSC_SCALAR_DOUBLE,
  PETSC_SCALAR_SINGLE,
  PETSC_SCALAR_LONG_DOUBLE,
  PETSC_SCALAR_HALF
} PetscScalarPrecision;

/*MC
   PetscAbs - Returns the absolute value of a number

   Synopsis:
   #include <petscmath.h>
   type PetscAbs(type v)

   Not Collective

   Input Parameter:
.  v - the number

   Level: beginner

   Note:
   The type can be integer or real floating point value, but cannot be complex

.seealso: `PetscAbsInt()`, `PetscAbsReal()`, `PetscAbsScalar()`, `PetscSign()`
M*/
#define PetscAbs(a) (((a) >= 0) ? (a) : (-(a)))

/*MC
   PetscSign - Returns the sign of a number as an integer of value -1, 0, or 1

   Synopsis:
   #include <petscmath.h>
   int PetscSign(type v)

   Not Collective

   Input Parameter:
.  v - the number

   Level: beginner

   Note:
   The type can be integer or real floating point value

.seealso: `PetscAbsInt()`, `PetscAbsReal()`, `PetscAbsScalar()`
M*/
#define PetscSign(a) (((a) >= 0) ? ((a) == 0 ? 0 : 1) : -1)

/*MC
   PetscMin - Returns minimum of two numbers

   Synopsis:
   #include <petscmath.h>
   type PetscMin(type v1,type v2)

   Not Collective

   Input Parameters:
+  v1 - first value to find minimum of
-  v2 - second value to find minimum of

   Level: beginner

   Note:
   The type can be integer or floating point value, but cannot be complex

.seealso: `PetscMax()`, `PetscClipInterval()`, `PetscAbsInt()`, `PetscAbsReal()`, `PetscSqr()`
M*/
#define PetscMin(a, b) (((a) < (b)) ? (a) : (b))

/*MC
   PetscMax - Returns maximum of two numbers

   Synopsis:
   #include <petscmath.h>
   type max PetscMax(type v1,type v2)

   Not Collective

   Input Parameters:
+  v1 - first value to find maximum of
-  v2 - second value to find maximum of

   Level: beginner

   Note:
   The type can be integer or floating point value

.seealso: `PetscMin()`, `PetscClipInterval()`, `PetscAbsInt()`, `PetscAbsReal()`, `PetscSqr()`
M*/
#define PetscMax(a, b) (((a) < (b)) ? (b) : (a))

/*MC
   PetscClipInterval - Returns a number clipped to be within an interval

   Synopsis:
   #include <petscmath.h>
   type clip PetscClipInterval(type x,type a,type b)

   Not Collective

   Input Parameters:
+  x - value to use if within interval [a,b]
.  a - lower end of interval
-  b - upper end of interval

   Level: beginner

   Note:
   The type can be integer or floating point value

   Example\:
.vb
  PetscInt c = PetscClipInterval(5, 2, 3); // the value of c is 3
  PetscInt c = PetscClipInterval(5, 2, 6); // the value of c is 5
.ve

.seealso: `PetscMin()`, `PetscMax()`, `PetscAbsInt()`, `PetscAbsReal()`, `PetscSqr()`
M*/
#define PetscClipInterval(x, a, b) (PetscMax((a), PetscMin((x), (b))))

/*MC
   PetscAbsInt - Returns the absolute value of an integer

   Synopsis:
   #include <petscmath.h>
   int abs PetscAbsInt(int v1)

   Input Parameter:
.   v1 - the integer

   Level: beginner

.seealso: `PetscMax()`, `PetscMin()`, `PetscAbsReal()`, `PetscSqr()`
M*/
#define PetscAbsInt(a) (((a) < 0) ? (-(a)) : (a))

/*MC
   PetscAbsReal - Returns the absolute value of a real number

   Synopsis:
   #include <petscmath.h>
   Real abs PetscAbsReal(PetscReal v1)

   Input Parameter:
.   v1 - the `PetscReal` value

   Level: beginner

.seealso: `PetscReal`, `PetscMax()`, `PetscMin()`, `PetscAbsInt()`, `PetscSqr()`
M*/
#if defined(PETSC_USE_REAL_SINGLE)
  #define PetscAbsReal(a) fabsf(a)
#elif defined(PETSC_USE_REAL_DOUBLE)
  #define PetscAbsReal(a) fabs(a)
#elif defined(PETSC_USE_REAL___FLOAT128)
  #define PetscAbsReal(a) fabsq(a)
#elif defined(PETSC_USE_REAL___FP16)
  #define PetscAbsReal(a) fabsf(a)
#endif

/*MC
   PetscSqr - Returns the square of a number

   Synopsis:
   #include <petscmath.h>
   type sqr PetscSqr(type v1)

   Not Collective

   Input Parameter:
.   v1 - the value

   Level: beginner

   Note:
   The type can be integer, floating point, or complex floating point

.seealso: `PetscMax()`, `PetscMin()`, `PetscAbsInt()`, `PetscAbsReal()`
M*/
#define PetscSqr(a) ((a) * (a))

/*MC
   PetscRealConstant - a compile time macro that ensures a given constant real number is properly represented in the configured
   precision of `PetscReal` be it half, single, double or 128-bit representation

   Synopsis:
   #include <petscmath.h>
   PetscReal PetscRealConstant(real_number)

   Not Collective

   Input Parameter:
.   v1 - the real number, for example 1.5

   Level: beginner

   Note:
   For example, if PETSc is configured with `--with-precision=__float128` and one writes
.vb
   PetscReal d = 1.5;
.ve
   the result is 1.5 in double precision extended to 128 bit representation, meaning it is very far from the correct value. Hence, one should write
.vb
   PetscReal d = PetscRealConstant(1.5);
.ve

.seealso: `PetscReal`
M*/
#if defined(PETSC_USE_REAL_SINGLE)
  #define PetscRealConstant(constant) constant##F
#elif defined(PETSC_USE_REAL_DOUBLE)
  #define PetscRealConstant(constant) constant
#elif defined(PETSC_USE_REAL___FLOAT128)
  #define PetscRealConstant(constant) constant##Q
#elif defined(PETSC_USE_REAL___FP16)
  #define PetscRealConstant(constant) constant##F
#endif

/*
     Basic constants
*/
/*MC
  PETSC_PI - the value of $ \pi$ to the correct precision of `PetscReal`.

  Level: beginner

.seealso: `PetscReal`, `PETSC_PHI`, `PETSC_SQRT2`
M*/

/*MC
  PETSC_PHI - the value of $ \phi$, the Golden Ratio, to the correct precision of `PetscReal`.

  Level: beginner

.seealso: `PetscReal`, `PETSC_PI`, `PETSC_SQRT2`
M*/

/*MC
  PETSC_SQRT2 - the value of $ \sqrt{2} $ to the correct precision of `PetscReal`.

  Level: beginner

.seealso: `PetscReal`, `PETSC_PI`, `PETSC_PHI`
M*/

#define PETSC_PI    PetscRealConstant(3.1415926535897932384626433832795029)
#define PETSC_PHI   PetscRealConstant(1.6180339887498948482045868343656381)
#define PETSC_SQRT2 PetscRealConstant(1.4142135623730950488016887242096981)

/*MC
  PETSC_MAX_REAL - the largest real value that can be stored in a `PetscReal`

  Level: beginner

.seealso: `PETSC_MIN_REAL`, `PETSC_REAL_MIN`, `PETSC_MACHINE_EPSILON`, `PETSC_SQRT_MACHINE_EPSILON`, `PETSC_SMALL`
M*/

/*MC
  PETSC_MIN_REAL - the smallest real value that can be stored in a `PetscReal`, generally this is - `PETSC_MAX_REAL`

  Level: beginner

.seealso `PETSC_MAX_REAL`, `PETSC_REAL_MIN`, `PETSC_MACHINE_EPSILON`, `PETSC_SQRT_MACHINE_EPSILON`, `PETSC_SMALL`
M*/

/*MC
  PETSC_REAL_MIN - the smallest positive normalized real value that can be stored in a `PetscReal`.

  Level: beginner

  Note:
  See <https://en.wikipedia.org/wiki/Subnormal_number> for a discussion of normalized and subnormal floating point numbers

  Developer Note:
  The naming is confusing as there is both a `PETSC_REAL_MIN` and `PETSC_MIN_REAL` with different meanings.

.seealso `PETSC_MAX_REAL`, `PETSC_MIN_REAL`, `PETSC_MACHINE_EPSILON`, `PETSC_SQRT_MACHINE_EPSILON`, `PETSC_SMALL`
M*/

/*MC
  PETSC_MACHINE_EPSILON - the machine epsilon for the precision of `PetscReal`

  Level: beginner

  Note:
  See <https://en.wikipedia.org/wiki/Machine_epsilon>

.seealso `PETSC_MAX_REAL`, `PETSC_MIN_REAL`, `PETSC_REAL_MIN`, `PETSC_SQRT_MACHINE_EPSILON`, `PETSC_SMALL`
M*/

/*MC
  PETSC_SQRT_MACHINE_EPSILON - the square root of the machine epsilon for the precision of `PetscReal`

  Level: beginner

  Note:
  See `PETSC_MACHINE_EPSILON`

.seealso `PETSC_MAX_REAL`, `PETSC_MIN_REAL`, `PETSC_REAL_MIN`, `PETSC_MACHINE_EPSILON`, `PETSC_SMALL`
M*/

/*MC
  PETSC_SMALL - an arbitrary "small" number which depends on the precision of `PetscReal` used in some PETSc examples
  and in `PetscApproximateLTE()` and `PetscApproximateGTE()` to determine if a computation was successful.

  Level: beginner

  Note:
  See `PETSC_MACHINE_EPSILON`

.seealso `PetscApproximateLTE()`, `PetscApproximateGTE()`, `PETSC_MAX_REAL`, `PETSC_MIN_REAL`, `PETSC_REAL_MIN`, `PETSC_MACHINE_EPSILON`,
         `PETSC_SQRT_MACHINE_EPSILON`
M*/

#if defined(PETSC_USE_REAL_SINGLE)
  #define PETSC_MAX_REAL             3.40282346638528860e+38F
  #define PETSC_MIN_REAL             (-PETSC_MAX_REAL)
  #define PETSC_REAL_MIN             1.1754944e-38F
  #define PETSC_MACHINE_EPSILON      1.19209290e-07F
  #define PETSC_SQRT_MACHINE_EPSILON 3.45266983e-04F
  #define PETSC_SMALL                1.e-5F
#elif defined(PETSC_USE_REAL_DOUBLE)
  #define PETSC_MAX_REAL             1.7976931348623157e+308
  #define PETSC_MIN_REAL             (-PETSC_MAX_REAL)
  #define PETSC_REAL_MIN             2.225073858507201e-308
  #define PETSC_MACHINE_EPSILON      2.2204460492503131e-16
  #define PETSC_SQRT_MACHINE_EPSILON 1.490116119384766e-08
  #define PETSC_SMALL                1.e-10
#elif defined(PETSC_USE_REAL___FLOAT128)
  #define PETSC_MAX_REAL             FLT128_MAX
  #define PETSC_MIN_REAL             (-FLT128_MAX)
  #define PETSC_REAL_MIN             FLT128_MIN
  #define PETSC_MACHINE_EPSILON      FLT128_EPSILON
  #define PETSC_SQRT_MACHINE_EPSILON 1.38777878078144567552953958511352539e-17Q
  #define PETSC_SMALL                1.e-20Q
#elif defined(PETSC_USE_REAL___FP16)
  #define PETSC_MAX_REAL             65504.0F
  #define PETSC_MIN_REAL             (-PETSC_MAX_REAL)
  #define PETSC_REAL_MIN             .00006103515625F
  #define PETSC_MACHINE_EPSILON      .0009765625F
  #define PETSC_SQRT_MACHINE_EPSILON .03125F
  #define PETSC_SMALL                5.e-3F
#endif

/*MC
  PETSC_INFINITY - a finite number that represents infinity for setting certain bounds in `Tao`

  Level: intermediate

  Note:
  This is not the IEEE infinity value

.seealso: `PETSC_NINFINITY`, `SNESVIGetVariableBounds()`, `SNESVISetComputeVariableBounds()`, `SNESVISetVariableBounds()`
M*/
#define PETSC_INFINITY (PETSC_MAX_REAL / 4)

/*MC
  PETSC_NINFINITY - a finite number that represents negative infinity for setting certain bounds in `Tao`

  Level: intermediate

  Note:
  This is not the negative IEEE infinity value

.seealso: `PETSC_INFINITY`, `SNESVIGetVariableBounds()`, `SNESVISetComputeVariableBounds()`, `SNESVISetVariableBounds()`
M*/
#define PETSC_NINFINITY (-PETSC_INFINITY)

PETSC_EXTERN PetscBool  PetscIsInfReal(PetscReal);
PETSC_EXTERN PetscBool  PetscIsNanReal(PetscReal);
PETSC_EXTERN PetscBool  PetscIsNormalReal(PetscReal);
static inline PetscBool PetscIsInfOrNanReal(PetscReal v)
{
  return PetscIsInfReal(v) || PetscIsNanReal(v) ? PETSC_TRUE : PETSC_FALSE;
}
static inline PetscBool PetscIsInfScalar(PetscScalar v)
{
  return PetscIsInfReal(PetscAbsScalar(v));
}
static inline PetscBool PetscIsNanScalar(PetscScalar v)
{
  return PetscIsNanReal(PetscAbsScalar(v));
}
static inline PetscBool PetscIsInfOrNanScalar(PetscScalar v)
{
  return PetscIsInfOrNanReal(PetscAbsScalar(v));
}
static inline PetscBool PetscIsNormalScalar(PetscScalar v)
{
  return PetscIsNormalReal(PetscAbsScalar(v));
}

PETSC_EXTERN PetscBool PetscIsCloseAtTol(PetscReal, PetscReal, PetscReal, PetscReal);
PETSC_EXTERN PetscBool PetscEqualReal(PetscReal, PetscReal);
PETSC_EXTERN PetscBool PetscEqualScalar(PetscScalar, PetscScalar);

/*@C
  PetscIsCloseAtTolScalar - Like `PetscIsCloseAtTol()` but for `PetscScalar`

  Input Parameters:
+ lhs  - The first number
. rhs  - The second number
. rtol - The relative tolerance
- atol - The absolute tolerance

  Level: beginner

  Note:
  This routine is equivalent to `PetscIsCloseAtTol()` when PETSc is configured without complex
  numbers.

.seealso: `PetscIsCloseAtTol()`
@*/
static inline PetscBool PetscIsCloseAtTolScalar(PetscScalar lhs, PetscScalar rhs, PetscReal rtol, PetscReal atol)
{
  PetscBool close = PetscIsCloseAtTol(PetscRealPart(lhs), PetscRealPart(rhs), rtol, atol);

  if (PetscDefined(USE_COMPLEX)) close = (PetscBool)(close && PetscIsCloseAtTol(PetscImaginaryPart(lhs), PetscImaginaryPart(rhs), rtol, atol));
  return close;
}

/*
    These macros are currently hardwired to match the regular data types, so there is no support for a different
    MatScalar from PetscScalar. We left the MatScalar in the source just in case we use it again.
 */
#define MPIU_MATSCALAR MPIU_SCALAR
typedef PetscScalar MatScalar;
typedef PetscReal   MatReal;

struct petsc_mpiu_2scalar {
  PetscScalar a, b;
};
PETSC_EXTERN MPI_Datatype MPIU_2SCALAR PETSC_ATTRIBUTE_MPI_TYPE_TAG_LAYOUT_COMPATIBLE(struct petsc_mpiu_2scalar);

/* MPI Datatypes for composite reductions */
struct petsc_mpiu_real_int {
  PetscReal v;
  PetscInt  i;
};

struct petsc_mpiu_scalar_int {
  PetscScalar v;
  PetscInt    i;
};

PETSC_EXTERN MPI_Datatype MPIU_REAL_INT PETSC_ATTRIBUTE_MPI_TYPE_TAG_LAYOUT_COMPATIBLE(struct petsc_mpiu_real_int);
PETSC_EXTERN MPI_Datatype MPIU_SCALAR_INT PETSC_ATTRIBUTE_MPI_TYPE_TAG_LAYOUT_COMPATIBLE(struct petsc_mpiu_scalar_int);

#if defined(PETSC_USE_64BIT_INDICES)
struct /* __attribute__((packed, aligned(alignof(PetscInt *)))) */ petsc_mpiu_2int {
  PetscInt a;
  PetscInt b;
};
struct __attribute__((packed)) petsc_mpiu_int_mpiint {
  PetscInt    a;
  PetscMPIInt b;
};
/*
 static_assert(sizeof(struct petsc_mpiu_2int) == 2 * sizeof(PetscInt), "");
 static_assert(alignof(struct petsc_mpiu_2int) == alignof(PetscInt *), "");
 static_assert(alignof(struct petsc_mpiu_2int) == alignof(PetscInt[2]), "");

 clang generates warnings that petsc_mpiu_2int is not layout compatible with PetscInt[2] or
 PetscInt *, even though (with everything else uncommented) both of the static_asserts above
 pass! So we just comment it out...
*/
PETSC_EXTERN MPI_Datatype MPIU_2INT /* PETSC_ATTRIBUTE_MPI_TYPE_TAG_LAYOUT_COMPATIBLE(struct petsc_mpiu_2int) */;
PETSC_EXTERN MPI_Datatype MPIU_INT_MPIINT /* PETSC_ATTRIBUTE_MPI_TYPE_TAG_LAYOUT_COMPATIBLE(struct petsc_mpiu_int_mpiint) */;
#else
  #define MPIU_2INT       MPI_2INT
  #define MPIU_INT_MPIINT MPI_2INT
#endif
PETSC_EXTERN MPI_Datatype MPI_4INT;
PETSC_EXTERN MPI_Datatype MPIU_4INT;

static inline PetscInt PetscPowInt(PetscInt base, PetscInt power)
{
  PetscInt result = 1;
  while (power) {
    if (power & 1) result *= base;
    power >>= 1;
    if (power) base *= base;
  }
  return result;
}

static inline PetscInt64 PetscPowInt64(PetscInt base, PetscInt power)
{
  PetscInt64 result = 1;
  while (power) {
    if (power & 1) result *= base;
    power >>= 1;
    if (power) base *= base;
  }
  return result;
}

static inline PetscReal PetscPowRealInt(PetscReal base, PetscInt power)
{
  PetscReal result = 1;
  if (power < 0) {
    power = -power;
    base  = ((PetscReal)1) / base;
  }
  while (power) {
    if (power & 1) result *= base;
    power >>= 1;
    if (power) base *= base;
  }
  return result;
}

static inline PetscScalar PetscPowScalarInt(PetscScalar base, PetscInt power)
{
  PetscScalar result = (PetscReal)1;
  if (power < 0) {
    power = -power;
    base  = ((PetscReal)1) / base;
  }
  while (power) {
    if (power & 1) result *= base;
    power >>= 1;
    if (power) base *= base;
  }
  return result;
}

static inline PetscScalar PetscPowScalarReal(PetscScalar base, PetscReal power)
{
  PetscScalar cpower = power;
  return PetscPowScalar(base, cpower);
}

/*MC
   PetscApproximateLTE - Performs a less than or equal to on a given constant with a fudge for floating point numbers

   Synopsis:
   #include <petscmath.h>
   bool PetscApproximateLTE(PetscReal x,constant float)

   Not Collective

   Input Parameters:
+   x - the variable
-   b - the constant float it is checking if `x` is less than or equal to

   Level: advanced

   Notes:
   The fudge factor is the value `PETSC_SMALL`

   The constant numerical value is automatically set to the appropriate precision of PETSc so can just be provided as, for example, 3.2

   This is used in several examples for setting initial conditions based on coordinate values that are computed with i*h that produces inexact
   floating point results.

   Example\:
.vb
  PetscReal x;
  if (PetscApproximateLTE(x, 3.2)) { // replaces if (x <= 3.2) {
.ve

.seealso: `PetscMax()`, `PetscMin()`, `PetscAbsInt()`, `PetscAbsReal()`, `PetscApproximateGTE()`
M*/
#define PetscApproximateLTE(x, b) ((x) <= (PetscRealConstant(b) + PETSC_SMALL))

/*MC
   PetscApproximateGTE - Performs a greater than or equal to on a given constant with a fudge for floating point numbers

   Synopsis:
   #include <petscmath.h>
   bool PetscApproximateGTE(PetscReal x,constant float)

   Not Collective

   Input Parameters:
+   x - the variable
-   b - the constant float it is checking if `x` is greater than or equal to

   Level: advanced

   Notes:
   The fudge factor is the value `PETSC_SMALL`

   The constant numerical value is automatically set to the appropriate precision of PETSc so can just be provided as, for example, 3.2

   This is used in several examples for setting initial conditions based on coordinate values that are computed with i*h that produces inexact
   floating point results.

   Example\:
.vb
  PetscReal x;
  if (PetscApproximateGTE(x, 3.2)) {  // replaces if (x >= 3.2) {
.ve

.seealso: `PetscMax()`, `PetscMin()`, `PetscAbsInt()`, `PetscAbsReal()`, `PetscApproximateLTE()`
M*/
#define PetscApproximateGTE(x, b) ((x) >= (PetscRealConstant(b) - PETSC_SMALL))

/*MC
   PetscCeilInt - Returns the ceiling of the quotation of two positive integers

   Synopsis:
   #include <petscmath.h>
   PetscInt PetscCeilInt(PetscInt x,PetscInt y)

   Not Collective

   Input Parameters:
+   x - the numerator
-   y - the denominator

   Level: advanced

  Example\:
.vb
  PetscInt n = PetscCeilInt(10, 3); // n has the value of 4
.ve

.seealso: `PetscMax()`, `PetscMin()`, `PetscAbsInt()`, `PetscAbsReal()`, `PetscApproximateLTE()`
M*/
#define PetscCeilInt(x, y) ((((PetscInt)(x)) / ((PetscInt)(y))) + ((((PetscInt)(x)) % ((PetscInt)(y))) ? 1 : 0))

#define PetscCeilInt64(x, y) ((((PetscInt64)(x)) / ((PetscInt64)(y))) + ((((PetscInt64)(x)) % ((PetscInt64)(y))) ? 1 : 0))

PETSC_EXTERN PetscErrorCode PetscLinearRegression(PetscInt, const PetscReal[], const PetscReal[], PetscReal *, PetscReal *);