function [pt_SA_SA, pt_SA_CT, pt_CT_CT] = gsw_pt_second_derivatives(SA,CT)

% gsw_pt_second_derivatives second derivatives of potential temperature
% =========================================================================
%
% USAGE:
% [pt_SA_SA, pt_SA_CT, pt_CT_CT] = gsw_pt_second_derivatives(SA,CT)
%
% DESCRIPTION:
% Calculates the following three second-order derivatives of potential
% temperature (the regular potential temperature which has a reference
% sea pressure of 0 dbar),
% (1) pt_SA_SA, the second derivative with respect to Absolute Salinity
% at constant Conservative Temperature,
% (2) pt_SA_CT, the derivative with respect to Conservative Temperature
% and Absolute Salinity, and
% (3) pt_CT_CT, the second derivative with respect to Conservative
% Temperature at constant Absolute Salinity.
%
% INPUT:
% SA = Absolute Salinity [ g/kg ]
% CT = Conservative Temperature (ITS-90) [ deg C ]
%
% SA & CT need to have the same dimensions.
%
% OUTPUT:
% pt_SA_SA = The second derivative of potential temperature (the
% regular potential temperature which has reference sea
% pressure of 0 dbar) with respect to Absolute Salinity
% at constant Conservative Temperature.
% pt_SA_SA has units of: [ K/((g/kg)^2) ]
% pt_SA_CT = The derivative of potential temperature with respect
% to Absolute Salinity and Conservative Temperature.
% pt_SA_CT has units of: [ 1/(g/kg) ]
% pt_CT_CT = The second derivative of potential temperature (the
% regular one with p_ref = 0 dbar) with respect to
% Conservative Temperature at constant SA.
% pt_CT_CT has units of: [ 1/K ]
%
% AUTHOR:
% Trevor McDougall and Paul Barker. [ help@teos-10.org ]
%
% VERSION NUMBER: 3.01 (29th March, 2011)
% This function is unchanged from version 2.0 (24th September, 2010).
%
% REFERENCES:
% IOC, SCOR and IAPSO, 2010: The international thermodynamic equation of
% seawater - 2010: Calculation and use of thermodynamic properties.
% Intergovernmental Oceanographic Commission, Manuals and Guides No. 56,
% UNESCO (English), 196 pp. Available from http://www.TEOS-10.org.
% See Eqns. (A.12.9) and (A.12.10) of this TEOS-10 Manual.
%
% McDougall T.J., P.M. Barker, R. Feistel and D.R. Jackett, 2011: A
% computationally efficient 48-term expression for the density of
% seawater in terms of Conservative Temperature, and related properties
% of seawater. To be submitted to Ocean Science Discussions.
%
% This software is available from http://www.TEOS-10.org
%
%==========================================================================

%--------------------------------------------------------------------------
% Check variables and resize if necessary
%--------------------------------------------------------------------------

if ~(nargin == 2)
error('gsw_pt_second_derivatives: Requires two inputs')
end %if

if ~(nargout == 3)
error('gsw_pt_second_derivatives: Requires three outputs')
end %if

[ms,ns] = size(SA);
[mt,nt] = size(CT);

if (mt ~= ms | nt ~= ns)
error('gsw_pt_second_derivatives: SA and CT must have same dimensions')
end

if ms == 1
SA = SA.';
CT = CT.';
transposed = 1;
else
transposed = 0;
end

%--------------------------------------------------------------------------
% Start of the calculation
%--------------------------------------------------------------------------

dSA = 1e-3; % increment of Absolute Salinity is 0.001 g/kg.
SA_l = nan(size(SA));
inds_l = find(SA>=dSA);
if ~isempty(inds_l)
SA_l(inds_l) = SA(inds_l) - dSA;
end
inds_l = find(SA<dSA);
if ~isempty(inds_l)
SA_l(inds_l) = 0;
end
SA_u = SA + dSA;

[pt_SA_l, pt_CT_l] = gsw_pt_first_derivatives(SA_l,CT);
[pt_SA_u, pt_CT_u] = gsw_pt_first_derivatives(SA_u,CT);

pt_SA_SA = (pt_SA_u - pt_SA_l)./(SA_u - SA_l);
% pt_SA_CT = (pt_CT_u - pt_CT_l)./(SA_u - SA_l); % can calculate this either way;

dCT = 1e-2; % increment of Conservative Temperature is 0.01 degrees C;
CT_l = CT - dCT;
CT_u = CT + dCT;

[pt_SA_l, pt_CT_l] = gsw_pt_first_derivatives(SA,CT_l);
[pt_SA_u, pt_CT_u] = gsw_pt_first_derivatives(SA,CT_u);

pt_SA_CT = (pt_SA_u - pt_SA_l)./(CT_u - CT_l);
pt_CT_CT = (pt_CT_u - pt_CT_l)./(CT_u - CT_l);

if transposed
pt_SA_CT = pt_SA_CT.';
pt_CT_CT = pt_CT_CT.';
end

end