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math Module Reference
Data Types | |
| interface | abscmp |
| interface | flipv |
| interface | relcmp |
| interface | reord |
| interface | sort |
| interface | swap |
Functions/Subroutines | |
| pure logical function, public | sabscmp (x, y, tol) |
| Return single precision absolute comparison \( | x - y | < \epsilon \). | |
| pure logical function, public | dabscmp (x, y, tol) |
| Return double precision absolute comparison \( | x - y | < \epsilon \). | |
| pure logical function | srelcmp (x, y, eps) |
| Return single precision relative comparison \( | x - y |<= \epsilon*|y| \). | |
| pure logical function | drelcmp (x, y, eps) |
| Return double precision relative comparison \( | x - y |/|y| < \epsilon \). | |
| pure real(kind=rp) function, public | lambert_w0 (x, niter) |
| Approximate the principal real branch of the Lambert W function for non-negative real x. | |
| subroutine, public | rzero (a, n) |
| Zero a real vector. | |
| subroutine, public | izero (a, n) |
| Zero an integer vector. | |
| subroutine, public | row_zero (a, m, n, e) |
| Sets row e to 0 in matrix a. | |
| subroutine, public | rone (a, n) |
| Set all elements to one. | |
| subroutine, public | copy (a, b, n) |
| Copy a vector \( a = b \). | |
| subroutine, public | masked_copy_0 (a, b, mask, n, n_mask) |
| Copy a masked vector \( a(mask) = b(mask) \). | |
| subroutine, public | masked_copy (a, b, mask, n, n_mask) |
| Copy a masked vector \( a(mask) = b(mask) \). | |
| subroutine, public | masked_gather_copy_0 (a, b, mask, n, n_mask) |
| Gather a masked vector to reduced contigous vector \( a = b(mask) \). | |
| subroutine, public | face_masked_gather_copy_0 (a, b, mask, facet, lx, ly, lz, n_mask) |
| Gather values from a face-local SEM field to a reduced contiguous vector. | |
| subroutine, public | masked_gather_copy (a, b, mask, n, n_mask) |
| Gather a masked vector to reduced contigous vector \( a = b(mask) \). | |
| subroutine, public | masked_scatter_copy_0 (a, b, mask, n, n_mask) |
| Scatter a contigous vector to masked positions in a target array \( a(mask) = b \). | |
| subroutine, public | masked_scatter_copy (a, b, mask, n, n_mask) |
| Scatter a contigous vector to masked positions in a target array \( a(mask) = b \). | |
| subroutine, public | cfill_mask (a, c, n, mask, n_mask) |
| Fill a constant to a masked vector. \( a_i = c, for i in mask \). | |
| subroutine, public | cmult (a, c, n) |
| Multiplication by constant c \( a = c \cdot a \). | |
| subroutine, public | cmult2 (a, b, c, n) |
| Multiplication by constant c \( a = c \cdot b \). | |
| subroutine, public | cdiv (a, c, n) |
| Division of constant c by elements of a \( a = c / a \). | |
| subroutine, public | cdiv2 (a, b, c, n) |
| Division of constant c by elements of a \( a = c / b \). | |
| subroutine, public | cadd (a, s, n) |
| Add a scalar to vector \( a_i = a_i + s \). | |
| subroutine, public | cadd2 (a, b, s, n) |
| Add a scalar to vector \( a_i = b_i + s \). | |
| subroutine, public | cfill (a, c, n) |
| Set all elements to a constant c \( a = c \). | |
| subroutine, public | cwrap (a, min_val, max_val, n) |
| Wrap value around a range [min, max) | |
| real(kind=rp) function, public | glsum (a, n) |
| Sum a vector of length n. | |
| real(kind=rp) function, public | glmax (a, n) |
| Max of a vector of length n. | |
| real(kind=rp) function, public | glamax (a, n) |
| Max of the absolute value of a vector of length n. | |
| integer function, public | glimax (a, n) |
| Max of an integer vector of length n. | |
| real(kind=rp) function, public | glmin (a, n) |
| Min of a vector of length n. | |
| integer function, public | glimin (a, n) |
| Min of an integer vector of length n. | |
| subroutine, public | chsign (a, n) |
| Change sign of vector \( a = -a \). | |
| real(kind=rp) function, public | vlamax (vec, n) |
| maximum absolute value of a vector of length n, rank-local | |
| real(kind=rp) function, public | vlmax (vec, n) |
| maximum value of a vector of length n | |
| real(kind=rp) function, public | vlmin (vec, n) |
| minimun value of a vector of length n | |
| subroutine, public | invcol1 (a, n) |
| Invert a vector \( a = 1 / a \). | |
| subroutine, public | invcol3 (a, b, c, n) |
| Invert a vector \( a = b / c \). | |
| subroutine, public | invers2 (a, b, n) |
| Compute inverted vector \( a = 1 / b \). | |
| subroutine, public | vcross (u1, u2, u3, v1, v2, v3, w1, w2, w3, n) |
| Compute a cross product \( u = v \times w \) assuming vector components \( u = (u_1, u_2, u_3) \) etc. | |
| subroutine, public | vdot2 (dot, u1, u2, v1, v2, n) |
| Compute a dot product \( dot = u \cdot v \) (2-d version) assuming vector components \( u = (u_1, u_2, u_3) \) etc. | |
| subroutine, public | vdot3 (dot, u1, u2, u3, v1, v2, v3, n) |
| Compute a dot product \( dot = u \cdot v \) (3-d version) assuming vector components \( u = (u_1, u_2, u_3) \) etc. | |
| real(kind=rp) function, public | vlsc3 (u, v, w, n) |
| Compute multiplication sum \( dot = u \cdot v \cdot w \). | |
| real(kind=rp) function, public | vlsc2 (u, v, n) |
| Compute multiplication sum \( dot = u \cdot v \). | |
| subroutine, public | add2 (a, b, n) |
| Vector addition \( a = a + b \). | |
| subroutine, public | add3 (a, b, c, n) |
| Vector addition \( a = b + c \). | |
| subroutine, public | add4 (a, b, c, d, n) |
| Vector addition \( a = b + c + d\). | |
| subroutine, public | sub2 (a, b, n) |
| Vector substraction \( a = a - b \). | |
| subroutine, public | sub3 (a, b, c, n) |
| Vector subtraction \( a = b - c \). | |
| subroutine, public | add2s1 (a, b, c1, n) |
| Vector addition with scalar multiplication \( a = c_1 a + b \) (multiplication on first argument) | |
| subroutine, public | add2s2 (a, b, c1, n) |
| Vector addition with scalar multiplication \( a = a + c_1 b \) (multiplication on second argument) | |
| subroutine, public | addsqr2s2 (a, b, c1, n) |
| Returns \( a = a + c1 * (b * b )\). | |
| subroutine, public | invcol2 (a, b, n) |
| Vector division \( a = a / b \). | |
| subroutine, public | col2 (a, b, n) |
| Vector multiplication \( a = a \cdot b \). | |
| subroutine, public | col3 (a, b, c, n) |
| Vector multiplication with 3 vectors \( a = b \cdot c \). | |
| subroutine, public | subcol3 (a, b, c, n) |
| Returns \( a = a - b*c \). | |
| subroutine, public | add3s2 (a, b, c, c1, c2, n) |
| Returns \( a = c1 * b + c2 * c \). | |
| subroutine, public | add4s3 (a, b, c, d, c1, c2, c3, n) |
| Returns \( a = c1 * b + c2 * c + c3 * d \). | |
| subroutine, public | add5s4 (a, b, c, d, e, c1, c2, c3, c4, n) |
| Returns \( a = a + c1 * b + c2 * c + c3 * d + c4 * e\). | |
| subroutine, public | subcol4 (a, b, c, d, n) |
| Returns \( a = a - b*c*d \). | |
| subroutine, public | addcol3 (a, b, c, n) |
| Returns \( a = a + b*c \). | |
| subroutine, public | addcol4 (a, b, c, d, n) |
| Returns \( a = a + b*c*d \). | |
| subroutine, public | addcol3s2 (a, b, c, s, n) |
| Returns \( a = a + s(b*c) \). | |
| subroutine, public | ascol5 (a, b, c, d, e, n) |
| Returns \( a = b \dot c - d \cdot e \). | |
| subroutine, public | p_update (a, b, c, c1, c2, n) |
| Returns \( a = b \dot c1 ( a - c2 \cdot c )\). | |
| subroutine, public | x_update (a, b, c, c1, c2, n) |
| Returns \( a = b \dot c1 ( a - c2 \cdot c )\). | |
| real(kind=rp) function, public | glsc2 (a, b, n) |
| Weighted inner product \( a^T b \). | |
| real(kind=rp) function, public | glsc3 (a, b, c, n) |
| Weighted inner product \( a^T b c \). | |
| real(kind=rp) function, public | glsc4 (a, b, c, d, n) |
| real(kind=rp) function, public | glsubnorm (a, b, n) |
| Returns the norm of the difference of two vectors \( \sqrt{(a-b)^T (a-b)} \). | |
| subroutine | sortrp (a, ind, n) |
| Heap Sort for double precision arrays. | |
| subroutine | sorti4 (a, ind, n) |
| Heap Sort for single integer arrays. | |
| subroutine | swapdp (b, ind, n) |
| sort double precision array acording to ind vector | |
| subroutine | swapi4 (b, ind, n) |
| sort single integer array acording to ind vector | |
| subroutine | reorddp (b, ind, n) |
| reorder double precision array - inverse of swap | |
| subroutine | reordi4 (b, ind, n) |
| reorder single integer array - inverse of swap | |
| subroutine | flipvdp (b, ind, n) |
| Flip double precision vector b and ind. | |
| subroutine | flipvi4 (b, ind, n) |
| Flip single integer vector b and ind. | |
| subroutine, public | absval (a, n) |
| Take the absolute value of an array. | |
| subroutine, public | pwmax2 (a, b, n) |
| Point-wise maximum of two vectors \( a = \max(a, b) \). | |
| subroutine, public | pwmax3 (a, b, c, n) |
| Point-wise maximum of two vectors \( a = \max(b, c) \). | |
| subroutine, public | cpwmax2 (a, b, n) |
| Point-wise maximum of scalar and vector \( a = \max(a, b) \). | |
| subroutine, public | cpwmax3 (a, b, c, n) |
| Point-wise maximum of scalar and vector \( a = \max(b, c) \). | |
| subroutine, public | pwmin2 (a, b, n) |
| Point-wise minimum of two vectors \( a = \min(a, b) \). | |
| subroutine, public | pwmin3 (a, b, c, n) |
| Point-wise minimum of two vectors \( a = \min(b, c) \). | |
| subroutine, public | cpwmin2 (a, b, n) |
| Point-wise minimum of scalar and vector \( a = \min(a, b) \). | |
| subroutine, public | cpwmin3 (a, b, c, n) |
| Point-wise minimum of scalar and vector \( a = \min(b, c) \). | |
| real(rp) function, dimension(3, 3), public | matinv39 (a11, a12, a13, a21, a22, a23, a31, a32, a33) |
| real(kind=xp) function, dimension(3, 3), public | matinv3 (a) |
| Performs a direct calculation of the inverse of a 3×3 matrix. M33INV and M44INV by David G. Simpson pure function version from https://fortranwiki.org/fortran/show/Matrix+inversion Invert 3x3 matrix. | |
| real(kind=rp) function, public | math_stepf (x) |
| Smooth step function S(x) Returns 0 for x <= 0, 1 for x >= 1, and smooth transition in between. | |
| real(kind=rp) function, public | math_dstepf (x) |
| Derivative of math_stepf with respect to x: d(stepf)/dx. | |
| subroutine, public | sqrt_inplace (a, n) |
| Sqrt a vector \( a = sqrt(a) \). | |
| subroutine, public | power (ap, a, p, n) |
| Take the power of a vector \( a^p \). | |
| pure subroutine, public | eig_sym2 (a11, a22, a12, e1, e2) |
| Eigenvalues of a symmetric 2x2 matrix, descending. | |
| pure subroutine, public | eig_sym3 (a11, a22, a33, a12, a13, a23, e1, e2, e3) |
| Eigenvalues of a symmetric 3x3 matrix, descending. | |
Variables | |
| real(kind=rp), parameter, public | neko_eps = epsilon(1.0_rp) |
| Machine epsilon \( \epsilon \). | |
| real(kind=dp), parameter, public | neko_eps_dp = epsilon(1.0_dp) |
| real(kind=sp), parameter, public | neko_eps_sp = epsilon(1.0_sp) |
| real(kind=rp), parameter, public | neko_m_ln2 = log(2.0_rp) |
| \( ln(2) \) | |
| real(kind=rp), parameter, public | pi = 4._rp*atan(1._rp) |
| \( \pi \) | |
Function/Subroutine Documentation
◆ absval()
◆ add2()
◆ add2s1()
◆ add2s2()
◆ add3()
◆ add3s2()
◆ add4()
◆ add4s3()
| subroutine, public math::add4s3 | ( | real(kind=rp), dimension(n), intent(inout) | a, |
| real(kind=rp), dimension(n), intent(in) | b, | ||
| real(kind=rp), dimension(n), intent(in) | c, | ||
| real(kind=rp), dimension(n), intent(in) | d, | ||
| real(kind=rp), intent(in) | c1, | ||
| real(kind=rp), intent(in) | c2, | ||
| real(kind=rp), intent(in) | c3, | ||
| integer, intent(in) | n | ||
| ) |
◆ add5s4()
| subroutine, public math::add5s4 | ( | real(kind=rp), dimension(n), intent(inout) | a, |
| real(kind=rp), dimension(n), intent(in) | b, | ||
| real(kind=rp), dimension(n), intent(in) | c, | ||
| real(kind=rp), dimension(n), intent(in) | d, | ||
| real(kind=rp), dimension(n), intent(in) | e, | ||
| real(kind=rp), intent(in) | c1, | ||
| real(kind=rp), intent(in) | c2, | ||
| real(kind=rp), intent(in) | c3, | ||
| real(kind=rp), intent(in) | c4, | ||
| integer, intent(in) | n | ||
| ) |
◆ addcol3()
◆ addcol3s2()
◆ addcol4()
◆ addsqr2s2()
◆ ascol5()
◆ cadd()
◆ cadd2()
◆ cdiv()
◆ cdiv2()
◆ cfill()
◆ cfill_mask()
◆ chsign()
◆ cmult()
◆ cmult2()
◆ col2()
◆ col3()
◆ copy()
◆ cpwmax2()
◆ cpwmax3()
◆ cpwmin2()
◆ cpwmin3()
◆ cwrap()
◆ dabscmp()
◆ drelcmp()
◆ eig_sym2()
◆ eig_sym3()
| pure subroutine, public math::eig_sym3 | ( | real(kind=dp), intent(in) | a11, |
| real(kind=dp), intent(in) | a22, | ||
| real(kind=dp), intent(in) | a33, | ||
| real(kind=dp), intent(in) | a12, | ||
| real(kind=dp), intent(in) | a13, | ||
| real(kind=dp), intent(in) | a23, | ||
| real(kind=dp), intent(out) | e1, | ||
| real(kind=dp), intent(out) | e2, | ||
| real(kind=dp), intent(out) | e3 | ||
| ) |
Closed form via the characteristic polynomial of the deviatoric part, which avoids an iterative solve for what is a per quadrature point diagnostic. A diagonal matrix is handled by the general path and comes back exact; only the isotropic case is special, because the deviatoric scaling is singular there.
- Note
- The smallest root is a difference of terms of order \( \lambda_{max} \), so its relative accuracy degrades as \( \epsilon \kappa \) – about 1e-10 at the 1e6 condition numbers this is used to detect, which is ample for a threshold test but would not be if \( \kappa \) approached \( 1/\epsilon \). That is why this works in double regardless of what rp is: in a single precision build the same amplification against \( \epsilon_{sp} \) would put a 6% error on \( \kappa \) at 1e6 and lose it entirely near 1e7, exactly where the answer matters most.
- Both call sites in coef_metric_condition() pass a matrix scaled to unit magnitude: the geometric factors divided by their largest entry, and the same factors Jacobi scaled to a unit diagonal. The latter is exactly the identity for an orthogonal element, which is why the isotropic branch above is not merely defensive.
- Parameters
-
a11 Diagonal entry (1,1). a22 Diagonal entry (2,2). a33 Diagonal entry (3,3). a12 Off-diagonal entry (1,2). a13 Off-diagonal entry (1,3). a23 Off-diagonal entry (2,3). e1 The largest eigenvalue. e2 The intermediate eigenvalue. e3 The smallest eigenvalue.
◆ face_masked_gather_copy_0()
| subroutine, public math::face_masked_gather_copy_0 | ( | real(kind=rp), dimension(n_mask), intent(inout) | a, |
| real(kind=rp), dimension(:, :, :, :), intent(in) | b, | ||
| integer, dimension(0:n_mask), intent(in) | mask, | ||
| integer, dimension(0:n_mask), intent(in) | facet, | ||
| integer, intent(in) | lx, | ||
| integer, intent(in) | ly, | ||
| integer, intent(in) | lz, | ||
| integer, intent(in) | n_mask | ||
| ) |
- Parameters
-
a Destination array of size n_mask.b Source face-local array indexed as b(:, :, facet, element).mask Mask array of length n_mask + 1, wheremask(0) = n_mask.facet Facet ids associated with the masked points. lx Number of points in the first SEM direction. ly Number of points in the second SEM direction. lz Number of points in the third SEM direction. n_mask Size of the mask and destination arrays.
Definition at line 398 of file math.F90.
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◆ flipvdp()
◆ flipvi4()
◆ glamax()
◆ glimax()
◆ glimin()
◆ glmax()
◆ glmin()
◆ glsc2()
◆ glsc3()
◆ glsc4()
◆ glsubnorm()
◆ glsum()
◆ invcol1()
◆ invcol2()
◆ invcol3()
◆ invers2()
◆ izero()
◆ lambert_w0()
◆ masked_copy()
◆ masked_copy_0()
| subroutine, public math::masked_copy_0 | ( | real(kind=rp), dimension(n), intent(inout) | a, |
| real(kind=rp), dimension(n), intent(in) | b, | ||
| integer, dimension(0:n_mask) | mask, | ||
| integer, intent(in) | n, | ||
| integer, intent(in) | n_mask | ||
| ) |
- Parameters
-
a Destination array of size n.b Source array of size n.mask Mask array of length n_mask + 1, where mask(0) = n_maskthe length of the mask array.n Size of the arrays aandb.n_mask Size of the mask array mask.
Definition at line 325 of file math.F90.
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◆ masked_gather_copy()
| subroutine, public math::masked_gather_copy | ( | real(kind=rp), dimension(n_mask), intent(inout) | a, |
| real(kind=rp), dimension(n), intent(in) | b, | ||
| integer, dimension(n_mask) | mask, | ||
| integer, intent(in) | n, | ||
| integer, intent(in) | n_mask | ||
| ) |
- Parameters
-
a Destination array of size n_mask.b Source array of size n.mask Mask array of length n_mask. the length of the mask array. n Size of the array b.n_mask Size of the mask array maskanda.
- Note
- Assumes
n .ge. n_mask.
Definition at line 433 of file math.F90.
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◆ masked_gather_copy_0()
| subroutine, public math::masked_gather_copy_0 | ( | real(kind=rp), dimension(n_mask), intent(inout) | a, |
| real(kind=rp), dimension(n), intent(in) | b, | ||
| integer, dimension(0:n_mask) | mask, | ||
| integer, intent(in) | n, | ||
| integer, intent(in) | n_mask | ||
| ) |
- Parameters
-
a Destination array of size n_mask.b Source array of size n.mask Mask array of length n_mask + 1, where mask(0) = n_maskthe length of the mask array.n Size of the array b.n_mask Size of the mask array maskanda.
- Note
- Assumes
n .ge. n_mask.
Definition at line 373 of file math.F90.
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◆ masked_scatter_copy()
| subroutine, public math::masked_scatter_copy | ( | real(kind=rp), dimension(n), intent(inout) | a, |
| real(kind=rp), dimension(n_mask), intent(in) | b, | ||
| integer, dimension(n_mask) | mask, | ||
| integer, intent(in) | n, | ||
| integer, intent(in) | n_mask | ||
| ) |
- Parameters
-
a Destination array of size n.b Source array of size n_mask.mask Mask array of length n_mask + 1, where mask(0) = n_mask + 1the length of the mask array.n Size of the array maskand arrayb.m Size of the mask array a.
- Note
- Assumes
n .ge. n_mask.
Definition at line 483 of file math.F90.
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◆ masked_scatter_copy_0()
| subroutine, public math::masked_scatter_copy_0 | ( | real(kind=rp), dimension(n), intent(inout) | a, |
| real(kind=rp), dimension(n_mask), intent(in) | b, | ||
| integer, dimension(0:n_mask) | mask, | ||
| integer, intent(in) | n, | ||
| integer, intent(in) | n_mask | ||
| ) |
- Parameters
-
a Destination array of size n.b Source array of size n_mask.mask Mask array of length n_mask + 1, where mask(0) = n_mask + 1the length of the mask array.n Size of the array maskand arrayb.m Size of the mask array a.
- Note
- Assumes
n .ge. n_mask.
Definition at line 458 of file math.F90.
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◆ math_dstepf()
◆ math_stepf()
◆ matinv3()
◆ matinv39()
| real(rp) function, dimension(3,3), public math::matinv39 | ( | real(kind=rp), intent(in) | a11, |
| real(kind=rp), intent(in) | a12, | ||
| real(kind=rp), intent(in) | a13, | ||
| real(kind=rp), intent(in) | a21, | ||
| real(kind=rp), intent(in) | a22, | ||
| real(kind=rp), intent(in) | a23, | ||
| real(kind=rp), intent(in) | a31, | ||
| real(kind=rp), intent(in) | a32, | ||
| real(kind=rp), intent(in) | a33 | ||
| ) |
◆ p_update()
◆ power()
◆ pwmax2()
◆ pwmax3()
◆ pwmin2()
◆ pwmin3()
◆ reorddp()
◆ reordi4()
◆ rone()
◆ row_zero()
◆ rzero()
◆ sabscmp()
◆ sorti4()
◆ sortrp()
◆ sqrt_inplace()
◆ srelcmp()
◆ sub2()
◆ sub3()
◆ subcol3()
◆ subcol4()
◆ swapdp()
◆ swapi4()
◆ vcross()
| subroutine, public math::vcross | ( | real(kind=rp), dimension(n), intent(out) | u1, |
| real(kind=rp), dimension(n), intent(out) | u2, | ||
| real(kind=rp), dimension(n), intent(out) | u3, | ||
| real(kind=rp), dimension(n), intent(in) | v1, | ||
| real(kind=rp), dimension(n), intent(in) | v2, | ||
| real(kind=rp), dimension(n), intent(in) | v3, | ||
| real(kind=rp), dimension(n), intent(in) | w1, | ||
| real(kind=rp), dimension(n), intent(in) | w2, | ||
| real(kind=rp), dimension(n), intent(in) | w3, | ||
| integer, intent(in) | n | ||
| ) |
◆ vdot2()
◆ vdot3()
| subroutine, public math::vdot3 | ( | real(kind=rp), dimension(n), intent(out) | dot, |
| real(kind=rp), dimension(n), intent(in) | u1, | ||
| real(kind=rp), dimension(n), intent(in) | u2, | ||
| real(kind=rp), dimension(n), intent(in) | u3, | ||
| real(kind=rp), dimension(n), intent(in) | v1, | ||
| real(kind=rp), dimension(n), intent(in) | v2, | ||
| real(kind=rp), dimension(n), intent(in) | v3, | ||
| integer, intent(in) | n | ||
| ) |
